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MATHEMATICS A GOOD BEGINNING IS BACK WITH A REVOLUTIONARY NEW 7th EDITION
Mathematics A Good Beginning (MGB), a best-selling textbook, has been on the market for many years providing exemplary learning materials for thousands of college students. Recently,the authors have become alarmed at the HIGH PRICESstudents pay for our textbooks while at the same time dismayed by a decline in the quality of services provided by commercial publishers.
To rectify this situation, the authors have acquired all rights to the manuscript and are announcing a contemporary 7th Edition with an abundance of new features making the text a sure winner. The content has been streamlined and serious attention has been given to creating artwork that teaches. To top that off, we have created many free resources that extend the book's functionality.
The Common Core State Standards have been carefully explained and integrated.
CCSS Mathemtical Practices are explained!
Trajectories with expectations are provided and aligned with children's activities.
Pedagogical Content Knowledge (PCK) is illuminated for each topic using guiding research and field experiences.
An Instructional Blueprint is devised to help teachers translate the Common Core State Standards, the trajectories, and Pedagogical Content Knowledge into effective activities for children.
Many resources accompany this book, including study guides and reproducible activity resource sheets.
Our art work can't be beat! It teaches as well as decorates. Our current theme for this edition is tessellations and it will introduce you to many interesting characters.
MGB will sell for MUCH LESS than competitive texts. Yet its features are SUPERIOR !
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Mathematics Courses The objective of the department is that all students completing a major in mathematics shall be able to demonstrate:
a solid base of mathematical skills: symbol manipulation, model construction and interpretation, application of definitions and theorems to particular instances;
ability to apply mathematics to problem-solving in realistic situations;
facility in use of computer and calculator tools to support and extend analysis and presentation of mathematical work;
understanding of the nature of mathematics as a logical system and ability to develop and present valid mathematical arguments and proofs;
ability to research, organize, and deliver a presentation (oral and written) on a topic in mathematics.
An overriding goal of the department is to assist each student of mathematics in assessing his or her own interests, achievements, and potential. The students choosing to major in mathematics may design a program emphasizing pure or applied mathematics, statistics and actuarial science, preparation for teaching, and/or preparation for graduate work. A major or minor in mathematics also provides a valuable complement to a variety of other majors offered at Hastings College.
Transfer coursework into the Mathematics or Computer Science majors will be handled as follows: Students wishing to transfer a course for MTH 251 for credit towards the Mathematics or Mathematics Education major must pass a departmental test before receiving such credit. Students who wish to take CSC 250 must either pass CSC 150 at Hastings College, or pass the Hastings College CSC 150 test out exam, at the proficiency level.
Mathematics courses numbered below 150 will not be calculated in the major or minor cumulative grade point average.
MTH 100 General Mathematics — 3 hours This is an elective course designed to allow students who complete the CLEP Examination in General Mathematics the opportunity to receive credit. The Mathematics Department will review the test scores and the written essays according to college standards and make a credit recommendation to the Academic Dean and the Registrar.
MTH 110 Math Foundations-Numbers & Patterns — 3 hours A course designed for pre-service elementary teachers. Sets, Venn diagrams, whole numbers, integers, rational numbers, decimals, percentages, and numeration systems of other cultures and time periods are studied. Emphasis is placed on how these topics are learned and taught in the elementary schools. Open only to Elementary Education majors. Each fall term.
MTH 120 Math Foundations-Geometry & Logic — 3 hours A course designed for pre-service elementary teachers. Beginning geometry, congruence, symmetry, measurement, elementary probability, and descriptive statistics are studied. Emphasis is placed on how these topics are learned and taught in the elementary schools. Open only to Elementary Education majors. Each spring term.
MTH 140 Pre-Calculus — 4 hours A study of analytic geometry and functions (rational, trigonometric, logarithmic and exponential) and their graphs, for those students needing additional preparation prior to taking calculus. Prerequisite: Two years of high school algebra. Spring, odd-numbered years.
MTH 150 Calculus I — 4 hours The first course in the calculus sequence. Functions and their graphs, limits and continuity, derivatives and their applications, antiderivatives, and definite integrals. Prerequisite: high school algebra and trigonometry or MTH 140. Each fall term.
MTH 202 Discrete Mathematics — 3 hours An introduction to mathematical ways of thinking about discrete systems, and using them to model reality. Topics may include: counting principles, logic, circuits, theory of codes, shortest route and minimal spanning tree algorithms. Prerequisite: High school algebra. May be offered during interim term.
MTH 210 Introduction to Statistics — 4 hours A study of descriptive and inferential statistics, including analysis and presentation of data, basic probability, random variables and their distributions, statistical inference, estimation and hypothesis testing, regression and correlation analysis, and one-way ANOVA. Prerequisite: High school algebra. Each long term.
MTH 251 Calculus III — 4 hours The final course in the calculus sequence. Vectors and vector-valued functions, functions of two or more variables, applications of calculus to curves and surfaces in Euclidean three-space. Prerequisite: MTH 160 (with C or better). Each fall term.
MTH 302 Geometry — 3 hours A study of advanced topics in Euclidean geometry and a survey of topics in modern geometries, including finite geometries, the projective plane, and groups of transformations of the plane. Prerequisite: MTH 160 (with C or better). Interim, odd-numbered years. MTH 308 Logic, Sets and Methods of Proof — 3 hours Theory and practice of mathematical proof and its foundation in symbolic logic. Construction of proofs about sets, relations, functions, real numbers, and integers. Prerequisite: MTH 150 (with C or better). Each interim term.
MTH 310 Teaching Math Foundations: Numbers Patterns — 1 hour
MTH 320 Teaching Math Foundations: Geometry/Logic — 1 hour
MTH 340 Teaching Pre-Calculus — 1 hour Courses for prospective teachers of mathematics. Students will participate in all aspects of MTH 110, 120, or 140, respectively, and will assume responsibility for teaching at least one segment of the course under supervision of the instructor. Recommended for mathematics education students and elementary education majors seeking an emphasis in mathematics. Prerequisite: Permission of the instructor. Concurrent with MTH 110, 120, 140. MTH 313 Linear Algebra with Applications — 4 hours A study of systems of linear equations, matrices, determinants, vector spaces, and linear transformations. Prerequisites: MTH 150 (with C or better), MTH 308 recommended. Each spring term.
MTH 404 Real Analysis — 4 hours A rigorous development of properties of the real number system and functions of a real variable. Topics include limits, continuity, differentiation, Riemann integration, and number sequences. Prerequisite: MTH 308 (with C or better). Fall, even-numbered years.
MTH 406 Introduction to Complex Analysis — 4 hours A study of functions of a complex variable. Topics include properties of the complex field, analytic functions, integration, and the calculus of residues. Prerequisite: MTH 308 (with C or better). On demand.
MTH 474 Advanced Topics in Mathematics — 2 hours Seminar approach to one or more advanced mathematical topics, depending on faculty and/or students' interests. Topics which have been or may be covered include Real Analysis II, Abstract Algebra II, and Advanced Mathematical Statistics. Prerequisites: Advanced standing and permission of the department. On demand.
MTH 484 Senior Project in Mathematics — 1-3 hours Student will work with a faculty member to research a mathematical topic, and will make a public presentation of the results of the study during the semester in which credit is awarded. Prerequisites: Advanced standing and permission of the department.
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LabVIEW offers hundreds of native analysis functions and several native and third-party math nodes that you can use to incorporate text-based math scripts into your graphical code. Discover how to add inline analysis to your acquisition system, review the different options for text-based math in LabVIEW, and discuss in-depth how to use the LabVIEW MathScript RT Module.
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Hello, my name is Katerina Koutouvas. I am a mathematics teacher and I studied at Valparaiso University. I am glad that you have taken the time to come to my website and to get a little glance into my world. Throughout my entire college career, I have used technology in all of my classes from microsoft office to movie editors and even different math programs. Due to this constant practice, I feel very confident that I will be able to incorporate technology into my classroom since just like math technology is everywhere.
Throughout the year, I will update my page so that you, parents and students will be able to check for homework and notes, and even receive extra help.
If I can help in anyway by posting more information or making this site easier to understand, just let me know. This site is meant to be helpful and if you all have any ideas that would increase the help it provides, send them right over.
If you want to know more about me, take a look at my glog,
Precalculus
This year we are going to be learning about trigonometric ratios. In addition, we will be using information that was taught in Algebra I and II and Geometry by applying it to different types of problems. We also will deal with sequences and series and finding equations that deal with the pattern. Then, we will deal with functions and graphs which may be a little different than what the students are typically uses to. The functions will be polynomials, rational, exponential, and logarithmic. Therefore, the students will experience a broad variety. More importantly, some of these functions will use real life examples which I believe make math much more interesting than it already is. Lastly, we will be learning about limits and derivatives which is when the material slowly approaches Calculus. This material is not always the easiest to grasp, but I am hoping that the students enjoy it as much as I do in the end.
This year, we will review concepts that have been learned in the previous years, but we will go more in depth in areas like factoring and several different properties. We will deal with writing and graphing linear equations and inequalities, while also solving systems of these equations which will determine where teh lines intersect if they do. In addition, once linear equations are understood, we will continue on to to quadratic equations; hence, we will cover the ideas of powers and exponents, but do not be scared, I will do my best in helping you understand the material. Lastly, as i said, factoring will be used quite often with polynomials and solving quadratics, but we will also use two formulas that should be very common to you. The quadratic formula and Pythagorean theorem will become your best friends.
Algebra is the first step in understanding mathematics, so do not be scared to jump right in, you only learn when you try. Mistakes help us learn and if you ever need help understanding something feel free to ask because as you know others probably feel the same and just are also scared. This is not a scary class, and I hope you all enjoy Algebra as much as I do.
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Testimonial
Bridgeway Math Book 1 Math Foundations
Bridgeway Math Foundations is the first book in a remedial math course written specifically to be used as a homeschool independent study course. No teacher's guides, no extra work, no extra instruction needed. Instead, it provides step by step easy to follow instructions to the student and plenty of practice to ensure that they are really mastering each concept. 728795162764
Bridgeway Math Foundations covers the topics of: basic operations, adding, subtracting, multiplying, and dividing, using whole numbers, place value, estimation, fractions, least common denominators, mixed numbers, decimals, and word problems. This book guides students through the first half of the Bridgeway course and prepares them for Bridgeway Pre-Algebra.
Somehow the way this remedial homeschool math course presents the material works! Kids just get it and find that they are able to succeed as they move on to more difficult concepts. The order makes sense, the easy to understand teaching and instructions, the number of practice questions for each concept… All carefully put together to ensure success.
A terrific foundations or remedial math course for children in 7th grade all the way up to high school seniors in need of extra instruction in math. And it works.
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Preface
Preface
To the Instructor
If you are familiar with our current companion volume, Essentials of Precalculus with Calculus Previews, Fifth Edition, you may know that for our 3-hour semester course in precalculus mathematics at Loyola Marymount University we have long favored a short text covering only what we consider to be basic material necessary for the successful completion of a course in calculus; a text that would allow time for instructors to work with their students to focus on strengthening their algebraic, logarithmic, and trigonometric skills.
This longer text, Precalculus with Calculus Previews, Fifth Edition, is a recognition of the needs of those instructors whose course syllabus contains topics not covered in Essentials of Precalculus, have more class time to cover extra material, or simply prefer to design their own course by being able to choose from a wider variety of topics.
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Channels: Primary and Secondary Education Spanish audio. Russian audio.
Wolfram|Alpha generates answers to questions in real time by doing computations on its own vast internal knowledge base—making it a valuable resource for education. In this video, educators of all levels share advice on getting started with Wolfram|Alpha.
This video is part 1 of a panel discussion featuring teachers who use Wolfram technologies, including Wolfram|Alpha and Mathematica, in their classrooms. In this video, they highlight some of the successes and lessons learned.
This video is part 2 of a panel discussion featuring teachers who use Wolfram technologies, including Wolfram|Alpha and Mathematica, in their classrooms. In this video, they highlight some of the successes and lessons learned.
Mathematica has the world's most sophisticated and convenient mathematical typesetting technology. This includes TraditionalForm, which transforms a large group of expressions into their conventionally used mathematical notation. Learn more about using TraditionalForm in this screencast.
Mathematica has a collection of Assistant palettes that provide immediate point-and-click access to an extensive range of Mathematica capabilities. This screencast demonstrates how the palettes serve as convenient entry points for novice users, especially in education.
Mathematica offers an interactive classroom experience that helps students explore and grasp concepts. Topics covered in this screencast include getting started, interactivity, and cross-discipline uses.
Mathematica gives seamless immediate access to an ever-growing library of carefully curated and continually updated data. This screencast shows you how to access and utilize Mathematica's integrated data.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Japanese audio.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Spanish audio.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Portuguese audio.
Mathematica users around the world have created a variety of materials—from examples and Demonstrations to coursewares and projects—many of which you can start using immediately, even with no prior experience with Mathematica. Learn more in this screencast.
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Online Equation Editor
This equation editor opens in a pop-up window when you click on the link below. You can enter math characters, symbols or expressions by clicking on the icons provided. A snippet of code appears below. Then edit the code, and type your numbers or variables in it, or some additional text.
After you're done, you can save (download) the gif image to your computer or copy it to a document.
The code is Latex and is familiar to many of us who have used Latex before, but it is fairly intuitive. For example, when you want a fraction and you press the fraction button, you will see the code
\frac{a}{b}
Just type your numerator in place of a, and your denominator in place of b. Those can be expressions as well.
You can make fractions, exponents, subscripts, square roots and other roots, sums (sigma sign), products (pi sign), integrals and limits. A list of Greek characters is included. The symbol list is fairly basic and includes some basic operation and relation symbols, set theory symbols, and a few arrows.
You can also change the font and the font size and the background color.
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Sets, Relations and Functions: Sets and their Representations, Union, intersection and complements of sets, and their algebraic properties, Relations, equivalence relations, mappings, one-one, into and onto mappings, composition of mappings.
Complex Numbers: Complex numbers in the form a+ib and their representation in a plane. Argand diagram. Algebra of complex numbers, Modulus and Argument (or amplitude) of a complex number, square root of a complex number. Cube roots of unity, triangle inequality.
Matrices and Determinants: Determinants and matrices of order two and three, properties of determinants, Evaluation of determinants.
Area of triangles using determinants, Addition and multiplication of matrices, adjoint and inverse of matrix.
Test of consistency and solution of simultaneous linear equations using determinants and matrices.
Quadratic Equations: Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots; Symmetric functions of roots, equations reducible to quadratic equations – application to practical problems.
Permutations and Combinations: Fundamental principle of counting; Permutation as an arrangement and combination as selection, Meaning of P(n,r) and C(n,r). Simple applications.
Mathematical Induction and its Applications
Binomial Theorem and its Applications: Binomial Theorem for a positive integral index; general term and middle term; Binomial Theorem for any index. Properties of Binomial Co-efficients. Simple applications for approximations.
Integral as limit of a sum. Properties of definite integrals. Evaluation of definite integrals; Determining areas of the regions bounded by simple curves.
Differential Equations: Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables. Solution of homogeneous and linear differential equations, and those of the type d2y/dx2 = f(x).
Two Dimensional Geometry: Recall of Cartesian system of rectangular co-ordinates in a plane, distance formula, area of a triangle, condition for the collinearity of three points and section formula, centroid and in-centre of a triangle, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
The straight line and pair of straight lines: Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line Equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines, homogeneous equation of second degree in x and y, angle between pair of lines through the origin, combined equation of the bisectors of the angles between a pair of lines, condition for the general second degree equation to represent a pair of lines, point of intersection and angle between two lines.
Circles and Family of Circles: Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle in the parametric form, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to the circle, length of the tangent, equation of the tangent, equation of a family of circles through the intersection of two circles, condition for two intersecting circles to be orthogonal.
Conic Sections: Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point(s) of tangency.
Three Dimensional Geometry: Coordinates of a point in space, distance between two points; Section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation.
Equations of a line and a plane in different forms; intersection of a line and a plane, coplanar lines, equation of a sphere, its centre and radius. Diameter form of the equation of a sphere.
Vector Algebra: Vectors and Scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product. Application of vectors to plane geometry.
Measures of Central Tendency and Dispersion: Calculation of Mean, median and mode of grouped and ungrouped data. Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data.
Probability: Probability of an event, addition and multiplication theorems of probability and their applications; Conditional probability; Bayes' Theorem, Probability distribution of a random variate; Binomial and Poisson distributions and their properties.
Statics: Introduction, basic concepts and basic laws of mechanics, force, resultant of forces acting at a point, parallelogram law of forces, resolved parts of a force, Equilibrium of a particle under three concurrent forces, triangle law of forces and its converse, Lami's theorem and its converse, Two parallel forces, like and unlike parallel forces, couple and its moment.
Dynamics: Speed and velocity, average speed, instantaneous speed, acceleration and retardation, resultant of two velocities. Motion of a particle along a line, moving with constant acceleration. Motion under gravity. Laws of motion, Projectile motion.
Carbon family Group 15 elements – Nitrogen family Group 16 elements – Oxygen family Group 17 elements – Halogen family Group 18 elements – Noble gases and Hydrogen.
Transition Metals including Lanthanides: Electronic configuration: General characteristic properties, oxidation states of transition metals. First row transition metals and general properties of their compounds-oxides, halides and sulphides.
General properties of second and third row transition elements (Groupwise discussion). Preparation and reactions, properties and uses of Potassium dichromate and Potassium permanganate.
Inner Transition Elements: General discussion with special reference to oxidation states and lanthanide contraction.
Illustration with examples of Compounds having not more than three same or different functional groups/atoms. Isomerism – Structural and stereoisomerism (geometrical and optical). Chirality – Isomerism in Compounds having one and two chiral Centres.
Synthetic and Natural Polymers: Classification of Polymers, natural and synthetic polymers (with stress on their general methods of preparation) and important uses of the following : Teflon, PVC, Polystyrene, Nylon-66, terylene, Bakelite.
Chemical Energetics and Thermodynamics: Energy changes during a chemical reaction, Internal energy and Enthalpy, Internal energy and Enthalpy changes, Origin of Enthalpy change in a reaction, Hess's Law of constant heat summation, numericals based on these concepts.
Enthalpies of reactions(Enthalpy of neutralization, Enthalpy of combustion, Enthalpy of fusion and vaporization).
Sources of energy (conservation of energy sources and identification of alternative sources, pollution associated with consumption of fuels. The sun as the primary source).
First law of thermodynamics; Relation between Internal energy and Enthalpy, application of first law of thermodynamics.
Second law of thermodynamics: Entropy, Gibbs energy, Spontaneity of a chemical reaction, Gibbs energy change and chemical equilibrium, Gibbs energy available for useful work.
Rates of Chemical Reactions and Chemical Kinetics: Rate of reaction, Instantaneous rate of reaction and order of reaction. Factors affecting rates of reactions – factors affecting rate of collisions encountered between the reactant molecules, effect of temperature on the reaction rate, concept of activation energy, catalyst.
Effect of light on rates of reactions. Elementary reactions as steps to more complex reactions. How fast are chemical reactions
Rate law expression. Order of a reaction (with suitable examples). Units of rates and specific rate constants. Order of reaction and effect of concentration (study will be confined to first order only).
Water and hydrogen peroxide, structure of water molecule and its aggregates, physical and chemical properties of water, hard and soft water, water softening, hydrogen peroxide – preparation, properties, structure and uses.
Work, Energy and Power: Concept of work, energy and power. Energy – kinetic and potential. Conservation of energy and its applications, Elastic collisions in one and two dimensions. Different forms of energy.
Rotational Motion and Moment of Inertia: Centre of mass of a two-particle system. Centre of mass of a rigid body, general motion of a rigid body, nature of rotational motion, torque, angular momentum, its conservation and applications.
Moment of Inertia, parallel and perpendicular axes theorem, expression of moment of inertia for ring, disc and sphere.
Gravitation: Acceleration due to gravity, one and two-dimensional motion under gravity. Universal law of gravitation, variation in the acceleration due to gravity of the earth. Planetary motion, Kepler's laws, artificial satellite – geostationary satellite, gravitational potential energy near the surface of earth, gravitational potential and escape velocity.
Engineering Entrance Exam Question Bank CD
Solids and Fluids: Inter-atomic and Inter-molecular forces, states of matter.
Heat and Thermodynamics: ThermalElectrostatics: Electric charge – its unit and conservation, Coulomb's law, dielectric constant, electric field, lines of force, field due to dipole and its behaviour in a uniform electric field, electric flux, Gauss's theorem and its applications.
Electric potential, potential due to a point charge. Conductors and insulators, distribution of charge on conductors. Capacitance, parallel plate capacitor, combination of capacitors, energy of capacitor.
Current Electricity: Electric current and its unit, sources of energy, cells- primary and secondary, grouping of cells resistance of different materials, temperature dependence, specific resistivity, Ohm's law, Kirchoff's law, series and parallel circuits.
Wheatstone Bridge with their applications and potentiometer with their applications.
Magnetic Effects of Currents: ORay Optics: Reflection and refraction of light at plane and curved surfaces, total internal reflection, optical fibre; deviation and dispersion of light by a prism; Lens formula, magnification and resolving power; microscope and telescope
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Limits : Its Methods & Applications
The presenter will teach Limits, their applications & methods of solving them in this session.
Many people often make a very critical mistake in math which is when solving something like y=(x²-4)/(x-2). We get the answer (x+2) and we say that f(2) is 4, which is not correct. When we solve such a fraction, we have an invisible condition saying that x?2. We can only say that y "approaches" 4 when x "approaches" 2.
Similarly, when we have y=tanx/sinx, people often infer that
y(0)=1/cos(0)=1 which is again incorrect. We have to consider that the original function is not defined at x=0 and hence, we need the limits to help us out in this case too.
We will learn the properties of limits & the methods of solving limits like the L Hôpital''s rule, the 1^(8) form, factorization & rationalization methods & just a brief introduction on the sandwich theorem as it is not in JEE''s syllabus.
Once we are clear with limits, continuity & differentiability will follow. They are a part of limits and will be discussed in the same session.
About Rishabh Dev (Teacher)
Rishabh Dev is a teacher in Calculus & other mathematics disciplines. Dev has a remarkable style of delivering the very advanced math concepts useful for the various competitive exams in a very basic manner understandable to all.
He has developed innovative ways of teaching mathematics to his students. Rishabh teaches online and also takes face-to-face math classes for students in Bangalore / Bengaluru Area, Karnataka
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This workshop is designed to provide an overview of exponential functions. Topics discussed include the general form of exponential functions, the initial value, change factors, growth and decay, compounding, and continuous compounding.
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Welcome to class. This is where you can get you daily classwork and assignments by clicking on the correct link below. I will also post handouts and review sheets under your specific class link from time to time. Please contact me by clicking on the e-mail link to the left or stop by rm. 65 (the portable by the football fields) if you have any questions about the class. I am available before and after school.
AP Calculus BC
Welcome to AP Calculus. This course covers differential, Integral, and Series Calculus. Students can receive credit for MTH 251, 252, 253 over the three terms.
Please see me as early as possible should problems arise. Remember that hotmath is not available for the calculus course.
This course applies mathematical concepts learned in previous math classes to various careers and situations. Management Science covers ways to look at routing problems used by delivery, communications, and transportation companies. Social Choice and Decision Making will look at election methods, fair division, and apportionment.
Game Theroy: The Mathematics of Competition looks at strategies for different types of games and situations that resemble them. Linear Programming deals with methods for optimizing production and profits. We will finish with Probability and where it applies.
This course is an extensive coverage of Euclidian geometry. Topics covered will be transformations and symmetry, angle relationships, similarity, right triangle trigonometry, special right triangles, laws of sine and cosine, proof, probability, constructions, polygons, 2-D dimensions and area, circles and chords, 3-D area and volume, constructions, and conic sections. This is a year long course and will be graded using a standards based system. Please see attachment.
Welcome to Pre-Calculus. This course replaces our college algebra and trigonometry classes and combines them into one year long class. In the first semester this course covers the algebra required for entry level math at the college level. Students may, if they wish receive, credit for MTH 111 in March (or Winter term at Chemeketa). The course covers advanced functions, logarithms, polynomial functions, rates, sequences, and series. The second semester covers trigonometry which is equivalent to MTH 112. Credit can be earned in June (spring term at Chemeketa). Topics include the radians and the unit circle, trigonometric functions and their graphs, identities, solving trig. equations, periodic fuctions, polar equations and complex numbers. A graphing calculator is required for much of the homework and classwork. A TI-84 plus is recommended but others will work. I won't be able to help with the programming and use of other brands of calculators. Please feel free to come in before or after school for additional help. The sooner you get help the more you will enjoy the course.
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Where do you need math, square roots, or algebra?
Students often wonder where in real life they would need any math skills. They do recognize the need for simple math, such as addition and multiplication, but in middle school, there are some topics where kids can start wondering why even study them (such as square roots or integers).
Then, in 8th or 9th grade, when students take algebra, many more can start asking this age-old question, "Where will I ever need algebra?"
The answer to that is that you need it in any occupational field that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade and commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You need algebra to take your SAT test or GED.
Algebra also lets you develop logical thinking and problem solving skills. It can increase your intelligence! (Actually, studying any math topic—even elementary math—can do that, if the mathematics is presented and taught in such a manner as to develop a person's thinking.)
You can admit to your student(s) that many mathematical concepts in algebra and beyond are not needed in every single occupation, especially in those of mostly manual labor. That is no big secret. Check Math Careers Database for the math requirements of 277 major occupations.
The website Algebra in the Real World has short movies, lesson guides, and student worksheets that show how algebra is used in with real word applications, such as roller coasters, banking, rice production, skyscrapers, solar power, and lots more.
Also, ask your students if they know for sure what they are going to do as adults. Most kids in middle school are not sure. If they are not sure, they'd better study algebra and learn all the math they can so that when they finally have some idea, they won't be stopped from a career because of not having studied algebra, geometry, or calculus.
And, even if students think they know what they want to be, how many times have young people changed their minds? Even we as adults don't necessarily know what kind of job or career changes are awaiting us. In times past, you could pretty well bank on either becoming a housewife (girl), or continuing in your father's occupation (boy). In today's world this is not so. Young people have more freedom in choosing - but the other side of that is that they need to study more to get a good basic education. Sometimes young people just need an adult to tell them that since they don't know all about their future, they need to keep studying, even math.
To futher help students see how mathematics and algebra are used in real world, check the free sample worksheets from Make It Real Learning activity books. These books focus on answering the question, "When am I ever going to use this?" and use REAL-LIFE data in the problems.
Another site to check out is Micron: Math in the Workplace, which contains a collection of real-world math problems and challenges contributed by a variety of businesses, demonstrating the relevance of math in today's world. Choose the "Math in the Worplace" tab.
Example: where do you need square roots?
As an example, let's say your students wonder, "Why do I need to know how to calculate the square root of a number? Are square roots really needed in life outside of math class?"
Here is one idea of how to show students one important real-life application of square root AND at the same time let them ponder where math is needed (and hopefully pique their interest into math problems in general). This lesson plan will work best when you have already taught the concept of square root but have not yet touched on the Pythagorean theorem.
Draw a square on board or paper, and draw one diagonal into it. Make the sides of the square to be, say, 5 units. Then make the picture to be a right triangle by wiping out the two sides of square. Then ask students how to find the length of the longest side of the triangle.
The students probably can't find the length if they haven't yet studied Pythagorean theorem. But that is part of the "game". Have you ever seen an advertisement where you couldn't tell what they were advertising? Then, later, in a few weeks the ad would change and reveal what it was all about. It makes you curious, doesn't it?
So, try to let them think about it for a few minutes and not tell them the answer. Hopefully it will pique their interest. Soon you will probably study the Pythagorean theorem anyway, since it often follows square root in the curriculum.
Then go on to the question: In what occupations or situations would you need to find the longest side of a right triangle if you know the two other sides? This can get them involved!
The answer is: in any kind of job that deals with triangles; for example, it is needful for carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers.
One time I observed people who needed to measure and mark on the ground where a building would go. Well, they had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a parallelogram. It is not easy to be sure that you have really drawn the two sides in a right angle.
Now, beyond this simple example, you need to understand the CONCEPT of square root in order to understand other math concepts. Studying math is like building a block wall or a building: you need the blocks on the lower part so you can build on them, and if you leave holes in your building, you can't build on the hole.
The concept of a square root is a prerequisite to, and ties in with, many other concepts in mathematics:
Comments
Instead of using the shotgun approach, schools should be teaching either professional or trade schools as appropriate at high school level. You could then teach specific math principles as appropriate. I am looking at my kids geometry, calc. classes and thinking what a waste of valuable teaching time. You would need higher math skills to figure out the billions of wasted dollars spent teaching upper math skills for no reason. What a shame.
truth you
I've always hated math but I never thought it would keep me from getting a college education. I found it so difficult to pass that I just quit. I still haven't graduated...
You need math so you can graduate high school and go to college.
Em
what kind of math skills do you need to be a construction worker
David Kutz
I think the best people to ask this about would be construction workers.... which I'm not. BUT I think construction workers would first of all need to know their geometry well, and everything about measuring and area and volume and such.
Then, you would probably need good grasp of percent and ratios... say maybe you're having to mix concrete, and you maybe need cement and sand and water in certain proportions in there...
And then, since construction work may involve all kinds of basic calculations, a construction worker probably needs to be able to do lots of mental math, and needs to be able to do rough estimates, as well as know how to do the exact calculations.
why do nurses study mathematics?
raizel magsalay
They need to know how to measure various things, understand metric system well with milliliters, milligrams, kilograms etc. They need to know how to calculate the right amount of medicine to give. Like for example, if you need to give 5mg of medicine per 10kg of body weight, then how much this person would need. Or, say 200mg of medicine as a tablet is equivalent to certain amount of the same stuff in liquid; then calculate how much is needed. They especially need to understand well decimal numbers and proportions.
I'm a bridge builder (carpenter) in San Diego, California who wishes I'd paid more attention in math class back when I was attending school. Every day now is a little bit of a math challenge. So in order to keep mt competitive edge in this high turn over industry I've desided to brush up on my math skills.
adrian chavira
what kind of maths do you need if you are a doctor? Is it the same as in nursing?
thanks
andy
Medical doctors need a solid understanding of chemistry to understand the workings of the human body and how medicines work, and for that, they need to know math well. Doctors also need logical thinking and be able to understand scientific writing and reasoning, and good math skills are essential for that as well.
All in all, to-be doctors should study all possible math courses in high school: algebra, geometry, trig, calculus, statistics.
what kind of
What jobs use pythagorean theorem? nessa
Check this link Jobs using Pythagorean Theorem from Math Careers Database. You can see it is various engineers, architects, surveyors, carpenters and other construction specialists, machinists, etc. Basically if you need triangles when designing things, then you need Pythagorean Theorem. Also if you're making big rectangles on land, such as when planning a building or farmland, Pythagorean Theorem is useful to know so you can check your 'rectangle' has right angles.
You'd be surprised at the level of mathematical expertise required in some "manual" jobs. I teach technical math at a community college, and constantly have students telling me they're using the trig and algebra concepts we're studying in class. One of the nicest things a student ever said to me is, "I do this stuff (meaning trig) in my machining class, but then I come here and I learn to understand it." When I taught technical math II, I was surprised at the sophistication of the course. Electrical technicians do lots of trig, vectors, complex numbers. The technical math sequence is not "easy". Never tell a student he won't need math in insert-profession-here. You just don't know.
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Book Details...
Basic Math and Pre-Algebra Workbook For Dummies
Get the confidence and the math skills you need to get started with algebra!
Are you about to jump into algebra? This easy-to-follow, hands-on workbook helps you brush up on your general math skills and practice the types of problems you'll encounter in your coursework. You get plenty of work space, step-by-step solutions to every problem, and expert tips on making the leap from arithmetic to algebra.
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Introduction To Discrete Mathematics
posted on: 24 May, 2012 | updated on: 25 May, 2012
There are two classifications of every experiment which are performed and measured according to the discrete values or continuous values as both have different perceptions. Introduction To Discrete Mathematics can be given as the branch of mathematics that refers to objects having only distinct and separate values or it can be said that this deals with the countable Sets. Continuous mathematics refers to branch of mathematics that shows the smoothly variation. Graph theory of hyper graphs, coding, the design of blocks, matroid theory, combinatorial and discrete Geometry, matrices etc. fields are covered by discrete mathematics.
Basically while giving discrete mathematics introduction one can not get the values on all the points as the function could be defined on some particular values only while continuous mathematics shows the combined result of smooth variation.
The introduction discrete mathematics topics listed above and the topics of number system like congruence and recurrence relation are also the part of discrete mathematics. To have a better understanding of these topics the study of algorithms is must.
The Set of objects in discrete mathematics can have infinite of finite Numbers. The removal of the errors is most significant in the discrete mathematics as compared with the continuous mathematics.
In computer science the areas of discrete mathematics are drawn on graph theory and logic. This is the study of algorithms for the getting the mathematical results. Besides that in logic, combinatorics, probability, algebra, geometry in all the sections discrete mathematics works.
Discrete is just the reverse process of continuous and the subject of engineering digital signal processing and signal systems are totally based on this concept. In discrete mathematics the values are measured as desperate values, while in continuous measurement is totally based on that smooth curve.
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Careers in Mathematics
Below are some links to websites that contain information on careers
in mathematics. This information
may be especially helpful to teachers
attempting to underscore the importance and usefulness of success
in
mathematics.
These sites have links to pages that contain career profiles for recent bachelors-level graduates in the mathematical sciences, job and graduate school information for mathematics majors, career information for high-school students, and pages devoted to non-academic career information. The sites also have several links to sites that have information about enrichment activities (contests, summer programs, etc.) and support programs (for example, tutoring sites) for undergraduate and high-school students.
This site gives information about the types of jobs available to statisticians, examples of the types of problems worked on by statisticians, the preparation (at high-school and college) needed to become a statistician, and profiles of famous statisticians.
This site has a wealth of information about programs available to women in mathematics. In addition to information about careers in mathematics the site highlights the accomplishments of female mathematicians and provides links to sites that feature enrichment programs, scholarships, grants, prizes and contests for female mathematicians.
The London Mathematical Society, The Royal Statistical Society and the Institute of Mathematics and its Applications sponsored this site. The goal of the site is to provide students aged 11 and up with information about the opportunities available in mathematics and statistics. The site has resources not only for students but also for teachers and parents.
This website gives information about the types of jobs, outside of the education field, available to students who study mathematics. Information is included about the preparation needed for the jobs, the work environment, and salary expectations.
SIAMís site has links that describe what an applied mathematician does, the type of environment in which an applied mathematician works, the preparation needed for a career as an applied mathematician and career profiles. The site also has information about job search resources and enrichment programs for students.
This site, developed for students (ages 8-18), their parents, their teachers, and their school counselors, serves as a portal about engineering and engineering careers. Students can ask an engineer or an undergraduate engineering student questions about this field of study. Teachers can find hands-on experiments and teaching resources.
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Search Course Communities:
Course Communities
Demos for Max-Min Problems
Instructors' notes break down steps for illustrating fundamental concepts for understanding and developing equations that model optimization problems, commonly referred to as max-min problems. The focus is on geometrically based problems so that animations can provide a foundation for developing insight and equations to model the problem. The common max-min problems illustrated include the following: "Maximize the area of a pen," "Minimize the time for rowing and walking," "Maximize the volume of an inscribed cylinder," "Maximize the area of an inscribed rectangle," "Determine the point on a curve closest to a fixed point," "Maximize the area for two pens," "Maximize the area of a rectangle inscribed in an isosceles triangle," "Maximize the printable region of a poster," "Construct a box of maximum volume," "Construct a cone of maximum volume," "Maximize the viewing angle of the Statue of Liberty," and "Minimize the travel time for light from one point to another."
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Algebra today, in the author's words \lq\lq plays a critical role in the development of the computer and communication technology that surround us in our daily lives." The goal of this undergraduate book is \lq\lq to show that the subject is alive, vibrant, exciting, and more relevant to modern technology than it has ever been." Accordingly the book presents the classical algebraic topics (modular arithmetic, groups, rings, fields and algebraic geometry) with a view on their applications to Computer Science, Error Correcting Codes and Cryptography and sections devoted to those applications are included through the book. Chapter 1 studies modular arithmetic with Section 1.8 devoted to bar codes and ISBN and Section 1.12 introducing the idea of public key cryptography. Chapter 2 provides the basic ideas about rings and fields, Section 2.14 studying error correcting codes (including BCH and Reed-Solomon codes). Chapters 3 and 4 deal with group theory and Section 4.10 illustrates the topic of substitution cipher with the description of the Enigma machine. Chapter 5 studies rings of polynomials, Gröbner bases and affine varieties. The last two chapters are devoted to elliptic curves. Chapter 6 introduces the concept of elliptic curve and its group law and states some classical results concerning elliptic curves over $\mathbb{B}$\, (theorems of Mordell-Weil, Mazur and Lutz-Nagell) and over a finite field (Hasse's theorem). Chapter 7 gathers some further topics related to elliptic curves as elliptic curve cryptosystems, the role of elliptic curves in the Wiles' proof of the Fermat last theorem or the Lenstra's factoring algorithm (the author also points out the existence of elliptic primality tests but he do not detail them). As conclusion, this book combines an introduction to abstract algebra with the presentation of some of its modern technological applications, which can contribute to awake up the interest of the students for this branch of the mathematics.
Reviewer:
Juan Tena Ayuso (Valladolid)
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Note to Readers: This is the original version of the Math Maturity page. On 10/6/09 it was renamed to Math Maturity v1 and a new Math Maturity page was created. While the two pages overlap considerably, the newer page has been heavily redesigned and edited. Most readers will likely decide the new version is superior to the older one.
"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, The Disciplined Mind: What All Students Should Understand, Simon & Schuster, 1999.)
"If we desire to form individuals capable of inventive thought and of helping the society of tomorrow to achieve progress, then it is clear that an education which is an active discovery of reality is superior to one that consists merely in providing the young with ready-made wills to will with and ready-made truths to know with…" (Jean Piaget; Swiss philosopher and natural scientist, well known for his work studying children, his theory of cognitive development; 1896–1980.)
Note to Readers
Math maturity is a complex topic. This article represents my explorations and understanding of the topic. It is a relatively long and convoluted article, drawing on a number of reference sources.
After reading the Introduction, you might want to jump to the Math Maturity section, thereby skipping much of the discussion of the research literature and more quickly arriving at some conclusions and recommendations.
Introduction
The three quotes given above help to give the flavor of this document. Math is a very important discipline both in its own right and because of its widespread applications in other disciplines.
Many people believe that the math education system in the United States is not nearly as effective as it should be. Over the years, there have been considerable efforts to improve the effectiveness of our math education system. Many of these efforts have focused on developing better curriculum and books, providing better preservice and inservice education for math teachers, requiring more years of math courses for precollege students, and setting more rigorous standards. There has also been a strong emphasis on encouraging women and minorities to take more math.
Current efforts to improve our math education system tend to be mostly focused on the same approaches. The general feeling seems to be that if we can just do more and better in these approaches, our math education system will improve.
Here are four important areas that have received much less attention.
There is substantial and mounting evidence that the math education curriculum in the United States is not designed and taught in a manner consistent with what is known about math cognitive development of students. Research in brain science is progressing more rapidly than our implementation of the results in our educational systems.
Our current math education system is not nearly as successful as we would like in helping students gain in their math creativity knowledge and skills, in their ability to attach and solve novel, complex, challenging problems, and in their ability to transfer their math knowledge and skills to problems outside the discipline of mathematics.
Our current math education system is still rather weak in teaching and learning in a manner that appropriately deals with forgetting. We know that students in math classes eventually (or, quite quickly) forget much of what they supposedly have learned. Although we spend quite a bit of time on review, we still face the constructivist problem that we are expecting students to build (construct) new knowledge on top of knowledge that they do not have.
In recent years, math educators have also had to deal with the steadily increasing capability and available of calculators and computers. In essence, we now need an education system that deals with both human brains and computer brains, and how to prepare humans to work effectively in environments where the computer capabilities increase significantly year by year.
There are many different approaches to the study of math education and in exploration of ways to improve our math eduction system. This document explores math intelligence, math cognitive development, and math maturity. It includes a focus on how to help students increase their level of math maturity.
In very brief summary, math maturity consists of an appropriate combination of math knowledge and skills, and the ability to think, understand, and solve problems using the math knowledge and skills. With disuse over time, one forgets much of their "learned" math knowledge and skills. However, one's level of math maturity—one's level of math-oriented thinking, understanding, and problem-solving—tends to have long term retention. Much more detail about math maturity is provided later in this document.
As with other documents in this IAE-pedia, the goal is to help improve education at all levels and throughout the world. Readers need to keep in mind that there are many different approaches to improving math education.
Marshmallows and Delayed Gratification
Quite a bit of formal education involves delayed gratification. This is certainly true in math education. When a student asks, "Why do I need to learn this?" a frequent response is, "You are going to need it next year." Of course, an common response nowadays is also, "It is going to be on the test." Personally, I find such a response rather unsatisfactory.
There has been some amusing and interesting research on a type of delayed gratification of young children. You can read a New Yorker magazine article on this, or view the short video on the test.
Youngsters are tested on whether they can delay eating a marshmallow (or some other "treat") in order to get two of the treats 15 minutes later. Only about 1/3 of the four-year old US children in the original research and 1/3 of the 4–6 year old Colombian children in research on children in that country were able to delay for 15 minutes.
Follow-up research on the US children 15 years later indicated that all who were able to delay their gratification for 15 minutes had been quite successful as students and in other parts of their lives.
Here is a math education quote from the New Yorker article:
Angela Lee Duckworth, an assistant professor of psychology at the University of Pennsylvania, is leading the program. She first grew interested in the subject after working as a high-school math teacher. "For the most part, it was an incredibly frustrating experience," she says. "I gradually became convinced that trying to teach a teen-ager algebra when they don't have self-control is a pretty futile exercise." And so, at the age of thirty-two, Duckworth decided to become a psychologist. One of her main research projects looked at the relationship between self-control and grade-point average. She found that the ability to delay gratification—eighth graders were given a choice between a dollar right away or two dollars the following week—was a far better predictor of academic performance than I.Q. She said that her study shows that "intelligence is really important, but it's still not as important as self-control." [Bold added for emphasis.]
Background on Innate Human Math Capabilities
There is research backing the idea that several month old human babies have innate ability to recognize small quantities, such as noticing that there is a difference between two of something and three of that thing. A variety of other animals have somewhat similar innate sense of quantity.
Although fractions are thought to be a difficult mathematical concept to learn, the adult brain encodes them automatically without conscious thought, according to new research in the April 8, 2009 issue of The Journal of Neuroscience. The study shows that cells in the intraparietal sulcus (IPS) and the prefrontal cortex - brain regions important for processing whole numbers - are tuned to respond to particular fractions. The findings suggest that adults have an intuitive understanding of fractions and may aid in the development of new teaching techniques.
"Fractions book The Math Gene (Devlin, 1999) presents an argument that the ability to learn to speak and understand a natural language such as English is a very strong indication that one can learn math.
In essence, Devin argues that a student's development of math knowledge and skills is mostly dependent on informal and formal education coming from parents, teachers, television, games, and so on. See his opening keynote presentation at the 2004 NCTM Annual Conference.
This insight helps us to understand one of the major challenges in our current math education system. A great many parents were not particularly successful in learning math, and typically they do not provide a "rich" math environment for their children. A great many elementary school teachers are not particularly strong in math. The math environments they provide in their classrooms tends to consist of "covering" the math book and its related curriculum. Their level of math maturity is modest, as is their interest in and enthusiasm for math.
AS a consequence of this, many young students do not gain nearly as high a level of math maturity as they might. This is not a consequence of their innate math abilities. Rather, it is a consequence of the informal and formal math education that they receive at home, in their community, and in their early years of schooling.
Intelligence and Intelligence Quotient (IQ)
"Did you mean to say that one man may acquire a thing easily, another with difficulty; a little learning will lead the one to discover a great deal; whereas the other, after much study and application, no sooner learns than he forgets?" (Plato, 4th century B.C.)
As the quote from Plato indicates, people have long been interested in intelligence. It has long been known thatpeople vary considerably in their rate and quality of their learning.
There is substantial research to support the contention that students of higher IQ learn faster and better than students of lower IQ. A teacher in a typical elementary school classroom may have one or two students who can learn twice as fast (and better) than the average students in the class, and one or two who learn half as fast (and not as well) as compared to the average students in the class.
Here is a little more recent history on measuring IQ. Quoting from the Wikipedia:
The Stanford-Binet test started with [the 1904 work of] the French psychologist Alfred Binet, whom the French government commissioned with developing a method of identifying intellectually deficient children for their placement in special education programs. As Binet indicated, case studies might be more detailed and helpful, but the time required to test many people would be excessive.
Later, Alfred Binet and physician Theodore Simon collaborated in studying mental retardation in French school children. Theodore Simon was a student of Binet's. Between 1905 and 1908, their research at a boys school, in Grange-aux-Belles, led to their developing the Binet-Simon tests; via increasingly difficult questions, the tests measured attention, memory, and verbal skill. Binet warned that such test scores should not be interpreted literally, because intelligence is plastic and that there was a margin of error inherent to the test (Fancher, 1985). The test consisted of 30 items ranging from the ability to touch one's nose or ear when asked to the ability to draw designs from memory and to define abstract concepts. Binet proposed that a child's intellectual ability increases with age. Therefore, he tested potential items and determined that age at which a typical child could answer them correctly. Thus, Binet developed the concept of mental age (MA), which is an individual's level of mental development relative to others.
A driving force in Binet's work and the work of others in the field of IQ is the goal of developing a measurement that is reasonably accurate in predicting future success in school, work, and other cognitive-related activities. For example, with accurate information one can better align formal schooling with the cognitive abilities of a student, and one can better advise a student about informal and formal academic and career choices.
Notice the initial measures of intelligence measured attention, memory, and verbal skill. An ADHD student might do poorly on such a test. A child growing up in a "rich" verbal environment will tend to score much better on such a test than a child growing up in a poorer oral communication environment.
Here are four important ideas that have been developed and/or more fully explored since the initial work of Binet:
A typical person's brain reaches full maturity at approximately 25 to 27 years of age. One's brain continues to change significantly over the years, showing a marked level of plasticity. Thus, continuing active use and education of one's brain can maintain and continue to improve its level of performance for a great many years after its full physical maturity is reached.
As Binet pointed out, a child's intellectual ability increases with age. This led researcher to "norm" the scores on intelligence tests, in an attempt to produce a number (called IQ) that remains relatively stable over time.
Many researchers have explored the idea of a single general intelligence factor called "g" versus multiple intelligences. There is a quite high level of correlation between "g" and the various multiple intelligences identified by Howard Gardner, Robert Sternberg, and other researchers who have focused on the general area of multiple intelligences.
In terms of the fourth point given above, logical/mathematical is one of the eight multiple intelligences identified by Gardner. Creativity is one of the three multiple intelligences identified by Sternberg. From the point of view of these two theories of multiple intelligences, learning to make effective use of one's innate math abilities and learning to make creative mathematical use of one's brain are ways to increase one's level of math maturity.
Measuring Intelligence
There are many different ways to attempt to measure intelligence. It turns out that this is a very challenging task. This is further complicated by the fact that intelligence is strongly influenced by both nature (one's genetic makeup) and nurture (informal and formal education and life experiences). The "nurture" component of intelligence is also affected by things like the quality of food one eats (starvation is bad for the brain), poisons (mercury and lead are bad for the brain), brain injuries (brain damage can severely disrupt a brain's capabilities), and so on.
Have you ever wondered why an average person has an IQ of about 100, and that this often does not change much over the years? Surely an average person develops quite a bit mentally as he or she grows from infancy to adulthood and learns a great deal during this time through informal and formal education and through life experiences.
The explanation to this situation lies in the way that intelligence is measured and reported. Measures of intelligence are usually normed in a manner that makes one's IQ a relatively stable number over the years. Historically this was done by dividing one's intelligence test score by one's chronological age. Researchers developed the idea of scaling intelligence test scores to produce a mean of 100 and a standard deviation of 15 (or 17, or 14, or …, depending on the people developing the test.)
Nowadays the scaling process is handled somewhat differently. An intelligence test is developed for a certain age range and group of people. The scores are then normed to produce a mean of 100 and a specified standard deviation such as 15 or some other number. Now, when a person in this age range takes this test, his or her IQ score is determined by looking up the test score in a table of values that converts test score to IQ by comparison with test scores of those used in the norming process.
Thus, for example, a person whose test score is close to the mean of the test scores used in the norming process will be assigned an IQ of approximately 100. Suppose that this person takes another IQ test ten years later. It may well be a different test with different questions, and designed to be suited to the person's current age. The score that he or she receives on the test will be compared to the scores achieved by people who were approximately this age when the test was created and normed. If his or her test score is near the mean of this norming group, the result will be an IQ of approximately 100.
Even with this norming process, IQ can change over time. As an example, consider a four year old who has grown up in extreme poverty and in a home and neighborhood environment that includes lead paint and other toxins, and a single parent who is holding down two jobs to make ends meet. Then the child's home environment changes markedly. Perhaps the single parent marries into greater wealth and the child now experiences a much better home and neighborhood environment. Moreover, the child goes to a high quality kindergarten and then on into high quality elementary school. Such changes can produce a marked increase in IQ.
Nature and Nurture
It is important to understand that intelligence depends on a combination of nature and nurture. On average, intelligence increases considerably as a person grows up, and it decreases as one grows old. It is the norming process in IQ that (artificially) makes it appear that one's intelligence is not changing over the years.
Here is a somewhat different way of looking at this question. A newborn with a healthy brain has a tremendous capacity to learn. The child's brain grows rapidly and learns rapidly. Just imagine the challenge of gaining oral fluency in one language. If the child happens to live in a bilingual or trilingual home and extended environment, the typical child will become bilingual or trilingual. Amazing! This represents a huge capacity to learn and to make use of one's learning.
My point is that the "average" person is very intelligent. Good informal and formal learning opportunities can greatly increase the Gc component of one's intelligence.
Studies of nature versus nurture are typically done making use of identical twins that were separated at birth. One can find varying results in the literature—with IQ being determined about 50 percent by nature all the way up to about 80 percent by nature, depending on the study.
Current research suggests that nature and nurture work together in a very complex manner, and that we have a long way to go in this area of research.
Multiple Intelligences
A human brain is a very complex organ. Many different parts and characteristics of a human brain contribute to its ability to learn and to deal with complex, challenging problems. Perhaps you have observed that some people seem to have more linguistic ability, or musical ability, or math ability that other people. Might there be significant discipline-oriented differences in intelligence? That is, might the differing characteristics of healthy brains have significant built-in inherent abilities to learn some disciplines better than others?
This question has led to a variety of multiple intelligences models and ongoing disagreements among proponents of these various models and proponents of a single factor model of intelligence. These are importatn disagreements. Suppose that nature can endow one person with the brain and hearing system to gain/have perfect pitch, another person to have much better than average propensity to develop very good spatial sense, and a third to have a propensity for developing much above average mathematical logic sense? If so, then perhaps educators would want to identify these varying characteristics in their students and better individualize programs of study to meet there varying students.
Howard Gardner is well known for his theory of Multiple Intelligences. His current theory includes eight different components, including logical/mathematical, and spatial. (Spatial intelligence is quite helpful in math as well as in other disciplines, such as art.)
Suppose that Howard Gardner's theory of multiple intelligences is essentially correct. Moreover, suppose that there are huge differences in the inherent and potential logical/mathematical abilities of students. If this is the case, then perhaps we are expecting many students in school to attempt to learn far more math than their brains are well suited to learn. Our school system's heavy emphasis on math puts many students at risk of not graduating from high school.
Personal comment: I am not tone deaf. However, my children like to label me as "tune" deaf. I tend to believe that I have somewhat below average inherent ability in music. It is not clear to me that I would have made it through high school if I had been required to take a stringent sequence of musical performance, composition, and theory courses in high school.
Howard Gardner based his theory of Multiple Intelligences partly on studies he did on brain damaged people. If ability in a particular area—such as spatial sense—is wiped out by damage to a particular part of one's brain, then this is taken as evidence that spatial is a distinct type of intelligence.
However, many people disagree with the work of Howard Gardner. One of the difficulties is that the various parts of one's brain work together, and that there is considerable plasticity.
To take a personal example, I (Dave Moursund) was quite good at math from my early childhood on, and I had little trouble in getting a doctorate in this discipline. It certainly helped that both my mother and father were high IQ people, and both were mathematicians. However, my spatial sense is terrible—well below average. At one time it was believe that a high spatial "IQ" was an essential part of being successful in math. More recent research indicates that many successful mathematicians have poor spatial sense. There is more than one way to look at a math problem!
Cognitive Development
Cognitive development is measured and studied in terms of a stage theory. Piaget is well known for the initial four-level stage theory that he developed. According to Piaget, a child moves from the Sensory Motor Stage to the Pre Operational Stage to the Concrete Operations Stage to the Formal Operations Stage.
The following chart is from the Piaget reference given above:
Quoting from the same reference:
Data from adolescent populations indicates only 30 to 35% of high school seniors attain the cognitive development stage of formal operations (Kuhn, Langer, Kohlberg & Haan, 1977). For formal operations, it appears that maturation establishes the basis, but a special environment is required for most adolescents and adults to attain this stage.
More modern versions of this stage theory have a much larger number of stages. See:
This article provides a 15-stage Piagetian-type model of cognitive development. Quoting from the article:
The acquisition of a new-stage behavior has been an important aspect of Piaget's theory of stage and stage change. Because of his controversial notions of stage and stage change, however, little research on these issues has taken place in the late twentieth century, at least among psychologists in the United States. The research that has taken place is being done by Neo-Piagetians. The neo-Piagetians more precisely defined stage, taking each of Piaget's substages and showing that they were in fact stages. In addition, three postformal stages have been added. Similar changes were made with Kohlberg's stages and substages. Commons, Richards, and Armon (1984) created a stage comparison table, comparing stage sequences from a number of different traditions, that stands today as the standard. This table shows that there is, essentially, only one stage sequence. Commons and Richards (1984a, b) presented their first General Stage Model at that time. Commons and colleagues (Commons, Trudeau, et al., 1998; Commons & Miller, 1998) later revised that model and expanded it downward, changing the name of the model to the Model of Hierarchical Complexity. Table 1a shows a complete list of the Orders of Hierarchical Complexity described in that model.
Researchers in cognitive development are faced by many of the same issues as researchers in IQ. Two of these issues are:
What (relative) roles do nature and nurture play in cognitive development?
Is cognitive development essentially domain independent or is does a theory of "multiple" cognitive developments better describe the field?
IQ and Cognitive Development are relatively closely related areas. (An IQ test and a cognitive development test may well make use of some of the same questions or activities.) A brain undergoes cognitive development from the time it first begins to form in a fetus. This cognitive development continues throughout life. (However, there can be cognitive decline due to age-related and other damage to the brain.)
IQ testing is one way to measure cognitive development. A test based on a stage theory of cognitive development is another approach to such measurement. Each of these measurement processes can be used to produce a number, such as an IQ number or a stage number.
The norming process in IQ measurement tends to produce a number that remains relatively stable over time. The stage measurement approach produces a stage level (number or designation) that increases over time as a person moves up a cognitive development scale.Relatively few people reach the top level of this 15-point scale.
Math Cognitive Development
There are a large number of hits on a Google search of math cognitive development. As an example, Stages of Math Development is a very short article that includes a math cognitive development stage theory model for children up to six years old.
Piaget did a lot of research in developing his 4-stage model of cognitive development. Besides his general interests in cognitive development, he also has a particular interest in math cognitive development. The following Dina and Pierre van Hiele geometry cognitive development scale was certainly inspired by Piaget's work. See also
Level
Name
Description
0
Visualization
Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).
1
Analysis
Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.
2
Informal Deduction
Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
3
Deduction
At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems, and formal proof is seen. The possibility of developing a proof in more than one way is seen. (Roughly corresponds to Formal Operations on the Piagetian Scale.)
4
Rigor
Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.
Notice that the van Hieles, being mathematicians, labeled their first stage Level 0. This is a common practice that mathematicians use when labeling the terms of a sequence. Piaget's cognitive development scale has four levels, numbers 1 to 4. The highest level in the van Hiele geometry cognitive development scale is one level above the highest level of the Piaget cognitive development scale.
The following scale was created (sort of from whole fabric) by
David Moursund. It represents his current insights into a six-level, Piagetian-type, math cognitive development scale.
Stage & Name
Math Cognitive Developments
Level 1. Piagetian and Math sensorimotor. Birth to age 2.
Infants use sensory and motor capabilities to explore and gain increasing understanding of their environments. Research on very young infants suggests some innate ability to deal with small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they can move to a target along a path requiring moving around obstacles, and can find their way back to a parent after having taken a turn into a room where they can no longer see the parent.
Level 2. Piagetian and Math preoperational. Age 2 to 7.
During the preoperational stage, children begin to use symbols, such as speech. They respond to objects and events according to how they appear to be. The children are making rapid progress in receptive and generative oral language. They accommodate to the language environments (including math as a language) they spend a lot of time in, so can easily become bilingual or trilingual in such environments.
During the preoperational stage, children learn some folk math and begin to develop an understanding of number line. They learn number words and to name the number of objects in a collection and how to count them, with the answer being the last number used in this counting process.
A majority of children discover or learn "counting on" and counting on from the larger quantity as a way to speed up counting of two or more sets of objects. Children gain increasing proficiency (speed, correctness, and understanding) in such counting activities.
In terms of nature and nurture in mathematical development, both are of considerable importance during the preoperational stage.
Level 3. Piagetian and Math concrete operations. Age 7 to 11.
During the concrete operations stage, children begin to think logically. In this stage, which is characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume, intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible).
While concrete objects are an important aspect of learning during this stage, children also begin to learn from words, language, and pictures/video, learning about objects that are not concretely available to them.
For the average child, the time span of concrete operations is approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as counting numbers) and concepts.
However, the level of abstraction in the written and oral math language quickly surpasses a student's previous math experience. That is, math learning tends to proceed in an environment in which the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and understanding of somewhat similar ideas that have already been acquired.
There is a substantial difference between developing general ideas and understanding of conservation of number, length, liquid, mass, weight, area, and volume, and learning the mathematics that corresponds to this. These tend to be relatively deep and abstract topics, although they can be taught in very concrete manners.
Level 4. Piagetian and Math formal operations. After age 11.
Starting at age 11 or 12, or so, thought begins to be systematic and abstract. In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts, problem solving, and gaining and using higher-order knowledge and skills.
Math maturity supports the understanding of and proficiency in math at the level of a high school math curriculum. Beginnings of understanding of math-type arguments and proof.
Piagetian and Math formal operations includes being able to recognize math aspects of problem situations in both math and non-math disciplines, convert these aspects into math problems (math modeling), and solve the resulting math problems if they are within the range of the math that one has studied. Such transfer of learning is a core aspect of Level 4.
Level 4 cognitive development can continue well into college, and most students never fully achieve Level 4 math cognitive development. (This is because of some combination of innate math ability and not pursuing cognitively demanding higher level math courses or equivalent levels on their own.)
Mathematical content proficiency and maturity at the level of contemporary math texts used at the upper division undergraduate level in strong programs, or first year graduate level in less strong programs. Good ability to learn math through some combination of reading required texts and other math literature, listening to lectures, participating in class discussions, studying on your own, studying in groups, and so on. Solve relatively high level math problems posed by others (such as in the text books and course assignments). Pose and solve problems at the level of one's math reading skills and knowledge. Follow the logic and arguments in mathematical proofs. Fill in details of proofs when steps are left out in textbooks and other representations of such proofs.
Level 6. Mathematician.
A very high level of mathematical proficiency and maturity. This includes speed, accuracy, and understanding in reading the research literature, writing research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.
Cognitive Acceleration In Mathematics Education
Shayer, Michael and Mundher, Adhami (June 2006). The Long-Term Effects from the Use of CAME (Cognitive Acceleration In Mathematics Education): Some Effects from the Use of the Same Principles in Y1&2, and the Maths Teaching of the Future. Proceedings of the British Society for Research into Learning Mathematics. Retrieved 8/14/09: Quoting from the article:
The CAME[1] project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive development of students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention. The RCPCM project[2], an intervention for the first two years of Primary education, doubled the proportion of 7 year-olds at the mature concrete level to 40%, with a mean effect-size of 0.38 S.D. on Key Stage 1 Maths. Yet, instead of the intervention intention, it is now suggested that a better view is to regard CAME as a constructive criticism of normal instructional teaching, with implications for the role of mathematics teachers and university staff in future professional development.
BACKGROUND TO CAME AND RCPCM
In the mid-70s CSMS[3] survey 14,000 children aged 10 to 16 were given three Piagetian tests to assess the range of thinking levels at each year. Figure 2 shows the findings.
By [age] 14 only 20% were showing formal operational thinking (3A&3B). This mattered at the time because current O-level science and maths courses, designed for grammar-school children in the top 20% of the ability range, required this level from the end of Y8. In the 80s the Graded Assessment in Maths scheme for the ILEA found that by the age of 12 the children's mathematics competence had a 12-year developmental gap between the above-average and those at what would later be National Curriculum Levels 1 and 2.
The CASE project shows the need to teach math in a manner that helps to increase cognitive development. Quoting from the article:
Enough evidence has been presented to show that the research has engendered class management skills in the teachers involved that realise Vygotsky's insistence that teaching should foster development as well as subject knowledge: that it should always aim ahead of where students presently are.
David Tall
David Tall is a Professor of Mathematical Thinking at the University of Warwick. Quoting from the linked site:
My interests in cognitive development in mathematics have matured over the years. Initially, as a mathematics lecturer at university, seeing mathematics through the eyes of a mathematician, I used mathematical theories (such as catastrophe theory) to formulate cognitive theory. After researching concepts of limits and infinity, I designed computer software to visualize calculus ideas using the idea of 'local straightness' rather than formal limits, and expanded my interest in visualization. Working first with Michael Thomas on algebra, then Eddie Gray on arithmetic, I was able to see the links between symbolism in arithmetic, algebra and calculus. This led directly to the theory of procepts which is concerned essentially with symbols that represent both process and concept and the ability to switch flexibly between processes 'to do' and concepts 'to think about'. Eddie and I were then able to see what we termed 'the proceptual divide '- the bifurcation between those children who remain focused on more specific procedures rather than develop the flexibility of proceptual thinking.
As. As, click here.
Stage Theory in Math
In a 4/18/09 email message "Michael Lamport Commons" <commons@tiac.net> wrote concerning a 15 point scale with the stages numbered 0–14:
What we have found is that to pass courses such as linear algebra, multivariate calculus and the like requires systematic stage 11 reasoning, one stage beyond formal stage 10. To pass abstract mathematics classes like modern algebra, topology, etc, when they are proof based, require metaystematic stage 12 reasoning. We also find that this is the earliest domain in which such reasoning develops.
A detailed discussion of the 15-level stage theory is available in the following reference:
Abstract: The Model of Hierarchical Complexity presents a framework for scoring reasoning stages in any domain as well as in any cross cultural setting. The scoring is based not upon the content or the participant material, but instead on the mathematical complexity of hierarchical organization of information. The participant's performance on a task of a given complexity represents the stage of developmental complexity. This paper presents an elaboration of the concepts underlying the Model of Hierarchical Complexity (MHC), the description of the stages, steps involved in universal stage transition, as well as examples of several scoring samples using the MHC as a scoring aid.
Michael Commons' work in this area is important for two reasons:
1. It emphasizes that a stage theory can be developed that cuts across domains and culture.
The term "postformal" has come to refer to various stage characterizations of behavior that are more complex than those behaviors found in Piaget's last stage—formal operations—and generally seen only in adults. Commons and Richards (1984a, 1894b) and Fischer (1980), among others, posited that such behaviors follow a single sequence, no matter the domain of the task e.g., social, interpersonal, moral, political, scientific, and so on.
Most postformal research was originally directed towards an understanding of development in one domain. The common approach to much of the work on postformal stages has been to specify a performance on tasks that develop out of those described by Piaget (1950, 1952) as formal-operational or out of tasks in related domains (e.g., moral reasoning). The assumption has been made that the predecessor task performances (formal operations), are in some way necessary to the development of their successor performances and proclivities (postformal operations). Unlike many of the other theories, the Model of Hierarchical Complexity (MHC), presented here (Commons, et al., 1998), generates one sequence that addresses all tasks in all domains and is based on a contentless, axiomatic theory.
This document summarizes the research on identifying and describing four Piagetian-type cognitive development stages that are above Piaget's Stage 4: Formal Operations. The assertion is that these four stages apply equally well in every discipline. The quote from Commons given earlier in this section provides an example of examining these higher-level stages within the discipline of mathematics.
IQ and Stage Theory
Email from Michael Commons to David Moursund 5/10/09 says:
The MHC [Model of Hierarchical Complexity] shows that stages are absolute and do not need in any way norms. Hierarchical Complexity is a major determinant of how difficult a task is. So stage and IQ should be quite correlated. My guess, is about an r of .5.
…
The evidence for stage change is a lot more clearly studied than IQ change. Most intervention buy 1 or 2 stages at the most. I know of no studies showing more.
The first quoted part reemphasizes that IQ measures are normed and Cognitive Development measures on a Piagetian-type stage scale are not. Commons suggests that IQ and Stage level are moderately correlated.
As noted earlier in this document, the norming process used to measure IQ tends to make the measure of one's IQ remain fairly stable over time. However, there are interventions that can increase IQ. It is not clear whether there are long term studies that indicate such interventions lead to long term increase in IQ.
Finally, Common's last statement above and general research on stage theory indicate that intervention can increase the rate that people move through stages, and that interventions can move the top stage level reached up one or two steps on a 15-stage measure.
Math Learning Disorders
ScienceDaily (Oct. 27, 2005) — ROCHESTER, Minn. -- In—skills that are essential for success at school, work and for coping with life in general. The results appear in the September-October issue of Ambulatory Pediatrics.
…
In the current study, Mayo Clinic researchers used different definitions of Math LD, analyzed school records of boys and girls enrolled in public and private schools in Rochester, Minn., and examined information from the students' medical records. They also looked at the extent to which Math LD occurs as an isolated learning disorder versus the extent to which it occurs simultaneously with Reading LD. This study is the first to measure the incidence—the occurrence of new cases—of Math LD by applying consistent criteria to a specific population over a long time. By considering the coexistence of Math LD and Reading LD across the students' entire educational experience (i.e., from grades K-12), the research presents a more comprehensive description of this association.
Examples
What is dyscalculia? Dyscalculia is a term referring to a wide range of life-long learning disabilities involving math. There is no single form of math disability, and difficulties vary from person to person and affect people differently in school and throughout life.
What are the effects of dyscalculia? Since disabilities involving math can be so different, the effects they have on a person's development can be just as different. For instance, a person who has trouble processing language will face different challenges in math than a person who has difficulty with visual - spatial relationships. Another person with trouble remembering facts and keeping a sequence of steps in order will have yet a different set of math-related challenges to overcome.
Here is some material quoted from an article by Jerome Schultz: The article is about central auditory processing disorders (CAPD) as it relates to learning math.
Let me take this opportunity to help our readers understand CAPD a bit better, since this condition often goes unrecognized or is misdiagnosed as ADHD. The American Speech-Language-Hearing Association (ASHA) established a task force in 1996 to gain a better understanding of central auditory processing disorders (CAPD) in children.
…
Learning multiplication tables involves auditory pattern recognition, and temporal factors (the order of the language). Differentiating 8 x 7 = 56 from 6 x 7 = 42 is very difficult, since these are abstract symbols for a particular quantity. If she just says them over and over again, she may remember one...until she hears the next one. Your daughter has to be instructed in a concrete visual, hands-on way to understand ("see" in her mind's eye) that a number represents a quantity. Otherwise, the times tables are just another jumble of numbers.
David Geary
David Geary is a highly prolific researcher in math cognitive development.
Here are some David Geary papers available on the Web.
David C. Geary, Mary K. Hoard, Jennifer Byrd-Craven, Lara Nugent, and Chattavee Numtee (July/August 2007). Cognitive Mechanisms Underlying Achievement Deficits in Children With Mathematical Learning Disability. Child Development. Retrieved 4/9/09. To access this paper go to find the paper in the list of papers, and click on its link. Quoting from this paper:
Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n=15) and low achieving (LA, n=44) were identified. These groups and a group of typically achieving (TA, n=46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M=6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed.
When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarily-novel advances influence one's ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturally-important knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for MLD. I begin in the first section with an organizing frame for approaching the task of decomposing the relation between evolved brain and cognitive systems and school-based learning and learning disability (LD). In the second section, I present a distinction between potentially evolved biologically-primary cognitive abilities and biologically-secondary abilities that emerge largely as a result of schooling (Geary, 1995), including an overview of primary mathematics. In the third section, I outline some of the cognitive and brain mechanisms that may be involved in modifying primary systems to create secondary abilities, and in the fourth section I provide examples of potential the sources of MLD based on the framework presented in the first section.
Geary, David (March 2006). Dyscalculia at an Early Age: Characteristics and Potential Influence on Socio-Emotional Development. Encyclopedia on Early Childhood Development. Retrieved 4/9/09: Quoting from the first part of this paper:
Introduction. Dyscalculia refers to a persistent difficulty in the learning or understanding of number concepts (e.g. 4 > 5), counting principles (e.g. cardinality – that the last word tag, such as "four," stands for the number of counted objects), or arithmetic (e.g. remembering that 2 + 3 = "5"). These difficulties are often called a mathematical disability. We cannot yet predict which preschool children will go on to have dyscalculia, but studies that will allow us to develop early screening measures are in progress. At this time and on the basis of normal development during the preschool years, it is likely that preschoolers who do not know basic number names, quantities associated with small numbers (< 4), how to count small sets of objects, or do not understand that subtraction results in less and addition results in more are at risk for dyscalculia.
Subject: How Common is Dyscalculia? Between 3 and 8% of school-aged children show persistent grade-to-grade difficulties in learning some aspects of number concepts, counting, arithmetic, or in related math areas. These and other studies indicate that these learning disabilities, or dyscalculia, are not related to intelligence, motivation or other factors that might influence learning. The finding that 3 to 8% of children have dyscalculia is misleading in some respects. This is because most of these children have specific deficits in one or a few areas, but often perform at grade level or better in other areas. About half of these children are also delayed in learning to read or have a reading disability, and many have attention deficit disorder.
There have only been a few large-scale studies of children with MD [Mathematical Disability] and all of these have focused on basic number and arithmetic skills. As a result, very little is known about the frequency of learning disabilities in other areas of mathematics, such as algebra and geometry. In any case, the studies in number and arithmetic are very consistent in their findings: Between 6 and 7% of school-age children show persistent, grade-to-grade, difficulties in learning some aspects of arithmetic or related areas (described below). These and other studies indicate that these learning disabilities are not related to IQ, motivation or other factors that might influence learning.
The finding that about 7% of children have some form of MD is misleading in some respects. This is because most of these children have specific deficits in one or a few subdomains of arithmetic or related areas (e.g., counting) and perform at grade-level or better in other areas of arithmetic and mathematics. The confusion results from the fact that standardized math achievement tests include many different types of items, such as number identification, counting, arithmetic, time telling, geometry, and so fourth. Because performance is averaged over many different types of items, some of which children with MD have difficulty on and some of which they do not, many of these children have standardized achievement test scores above the 7th percentile (though often below the 20th).
Math Maturity
The previous parts of this document have explored Math IQ and Math Cognitive Development. They provide a theoretical underpinning for the discussion of math maturity given in the remainder of this document.
Components of Math Maturity
The term math maturity is widely used by mathematicians and math educators. For example, a middle school teacher may say, "I don't think Pat has the necessary math maturity to take an algebra course right now." It is clear that the teacher is not talking about Pat's math content knowledge.
Probably Pat has completed the prerequisite coursework. Perhaps Pat is weak in math reasoning and thinking, tends to learn math by rote memorization, has little interest in math, and shows little persistence in working on challenging math problems. The teacher feels that with this background, Pat is apt to struggle in algebra and likely fail the course.
At the university level, the dominant component in the literature of math maturity is "proof" and the logical,critical, creative reasoning and thinking involved in understanding and doing proofs. A person with a high level of math maturity has studied math at a level that requires substantial understanding of proof and regular demonstration of the ability to do proofs.
K-12 math has only a modest emphasis on formal proofs. However, as students move up in the math curriculum, they face a growing challenge to make mathematical arguments that describe and justify the steps they take in solving challenging math problems. This is a type of logical/mathematical (proof) activity.
The following list contains some components of math maturity. An increasing level of math maturity is demonstrated by:
1. An increasing capacity in the logical, critical, creative reasoning and thinking involved in understanding and solving problems and in understanding and doing proofs.
2. An increasing capacity to move beyond rote memorization in recognizing, posing, representing, and solving math problems. This includes transfer of learning of one's math knowledge and skills to problems in many different disciplines.
3. An increasing capability to communicate effectively in the language and ideas of mathematics. This includes:
A. Mathematical speaking and listening fluency.
B. Mathematical reading and writing fluency.
C. Thinking and reasoning in the language and images of mathematics.
4. An increasing capacity to learn mathematics—to build upon one's current mathematical knowledge and to take increasing personal responsibility for this learning.
5. Improvements in other factors affecting math maturity such as attitude, interest, intrinsic motivation, focused attention, perseverance and delayed gratification, having math-oriented habits of mind, and acceptance of and fitting into the "culture" of the discipline of mathematics.
Email from Joseph Dalin 6/14/09
The following email message sent to David Moursund was in response to a email message about math and Talented & Gifted education sent to an National Council of Supervisors of Mathematics distribution list.
Hi,
The major question is: what is a Talented and Gifted persons or students? What is his/her special capabilities? Memorizing or the capability of understanding? Do the Talented and Gifted students learn differently? Do they understand symbolic abstract language better, or significantly better, than ordinary students?
The basic education of mathematical understanding and creative thinking is gained through solving problems of cases which deal within the child's environment, experience and conceptual system.
Algebra is a symbolic abstract nonhuman language. In order to understand such a language there is a need to:
Have a "mathematical thinking maturity" and than—to translate the symbolic representations into graphic representation and learn through self experience, exploration and discovery (which is the way human beings learn).
It can be achieved by comprehensive integration of Visual-dynamic-quantitative computer software into the teaching and learning process of school mathematics.
Such an approach should be applied for all students, not only Talented and Gifted students.
I don't believe that, in general, students of 5th grade have the "mathematical learning maturity" for learning algebra 1.
I don't believe that most students of 7th grade are capable to leaning algebra 1 through its symbolic representation only. Anyhow, what's the rush? Learning is a long journey….
I don't believe that most students, even in higher grades, are capable to really understand Algebra by learning only through its symbolic representations. That's the main reason of poor achievement, failure and frustration of school mathematics education which is based on teaching symbolic mathematics.
Three of the paragraphs have been bolded for extra emphasis. The message is that math maturity is a key issue in determining when a student (whether TAG or not) should begin an algebra course. The message also suggests that current widely used methods for teaching introductory algebra do not adequately address the challenge of learning to deal with a high level of abstractness and translating it into personally meaningful understanding.
Clyde Greeno
The following is quoted from an email message sent by Clyde Greeno to the National Council of Supervisors of Mathematics distribution list on 4/9/09:
The entrenched "developmental" algebra curriculum (like the HS algebra curriculum) is a direct decedent from SMSG's calculus-preparatory HS algebra—which has served more to filter students out of the mathematics curriculum than to empower them for success within it. [No wonder that educators now are concerned about the "Algebra 2 for everyone" movement among state legislatures.]
Extensive clinical research has revealed that the primary cause for students' difficulties with algebra is simply that algebra curricula within the SMSG lineage badly violate scientifically established principles of the developmental psychology of mathematical learning—i.e. of "mathematics as common sense". [The original SMSG version was created two decades before America began to understand Piaget.] Ironically, the resulting "developmental" algebra curriculum is anything but developmental. The coming reformation will be guided by psychology.
Clinical methods quickly reveal that students learn the usual essentials of algebra better, faster, and more easily through the context of functions. That is partly because the field of algebra really is all about the study of operations/functions—even the SMSG curriculum was covertly about functions—even though that context still is badly hidden by the current curriculum.
Education for Increasing Math Maturity
This section is a work in progress.
Math maturity increases over time through:
General overall increase in cognitive development.
Learning math in a manner that facilitates higher-order creative thinking, problem solving, theorem proving, communicating in and about math, and learning to learn math.
Working with math teachers who have a higher level of math maturity than oneself, and being taught at a level that is a little above one's current level of math maturity.
Math maturity is strongly affected by one's informal education, formal education, and life experiences. As one's brain grows and as one is engaged in informal and formal education, one's overall intelligence grows and one's level of cognitive development grows. If one's education and experiences have an appropriate math component, one's math maturity will increase.
Assessment of Math Maturity
This section is a Work in Progress and definitely needs input from a lot of people.
It is relatively easy to make use of the term math maturity and to claim it is an important concept or goal in math education. It is much more difficult to develop assessment instruments that can be used for self-assessment (by students), for assessment by people interested in measuring how well our math education system is doing in helping students to develop math maturity, and as an aid to student placement in courses.
Good math teachers are able to estimate the math maturity of their students through a one-on-one conversation, by listening to the breadth and depth of questions a student raises in class, by listening to the breadth and depth of answers a student gives to questions raised during a class, through analysis of a student's homework and test answers, and so on. There are many clues available in these information sources. However, it is a major challenge to identify them and teach less qualified teachers to learn to observe and make use of these information sources.
Good math teachers can determine if a student has the math maturity to effectively deal with the content the teacher wants to teach and whether a student is apt to be bored by the level and pace of what is to be taught.
With some practice, students can gain skill in self-assessing their level of math maturity and progress they are making in increasing their level of math maturity. Part of a useful approach is a self-assessment based on insights into learning by rote memory versus learning for understanding. Another is self-assessment on dealing with "challenging" problems that draw upon math covered a few weeks ago, much earlier in the school year, and in previous school years.
Still another approach is through self-assessment of how well one can explain to oneself and to others the thought processes and understanding used in attacking challenging problems and proofs. In this, however, one needs to be aware that people who are good at math often have intuitive or not readily explained leaps of insight. Such leaps often do not lend themselves to the "show your work" type of requirement that most teachers require of their students.
Computers and Math Maturity
"Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination." (Albert Einstein)
"My familiarity with various software programs is part of my intelligence if I have access to those tools." (David Perkins,1992.)
The two quotes capture the essence of this section. An intact human mind and body has tremendous capabilities. However, it also has severe limitations. Over many thousands of years humans have been developing tools that help to overcome some of these physical and mental limitations.
Thus, for example, we have developed telescopes for "far seeing" and microscopes for "near seeing" that far exceed the capabilities of the human visual system. We have developed reading, writing, and arithmetic that are wonderful aids to one's brain. We have developed machines such as cars, airplanes, and bulldozers. We have developed highly automated manufacturing facilities.
Now, we have Information and Communication Technology. It plays a role in many of our previously developed tools, and it provides a new type of intelligence. Machine intelligence (artificial intelligence) can be thought of as a new type of brain, or as an auxiliary brain.
In terms of the document you are now reading, the major question is the nature and extent to which the computer brain adds to the capabilities of human intelligence and human cognitive development. That is, as educators we now need to think in terms of nature, nurture, and machine intelligence.
Here is a concrete example. Spatial intelligence in one of the eight Multiple Intelligences on Howard Gardner's list. We have long had maps and compasses to help people deal with certain types of spatial problems. We now have computerized maps (for example, think in terms of Google Earth) and GPS systems that can aid in solving some of the spatial problems that people face. In essence, such tools increase the intelligence of their users.
David Tall's Work
The article Tall (2000) discusses the cognitive load that is inherent to learning and using mathematics. The cognitive load is reduced through learning the language and symbolism of math so that one can use it rapidly and accurately at a subconscious level—in the same way that one uses their native language in speaking and writing.
Calculators and computers can play a significant role in math education. Quoting from Tall (2000):
The development of symbol sense throughout the curriculum
faces a number of major re-constructions causing increasing
difficulties to more and more students as they are faced with
successive new ideas that require new coping mechanisms.
For many it leads to the satisfying immediate short-term needs
of passing examinations by rote-learning procedures. The
students may therefore satisfy the requirements of the current
course and the teacher of the course is seen to be successful.
If the long-term development of rich cognitive units is not set in
motion, short-term success may only lead to increasing
cognitive load and potential long-term failure.
…
Given the constraints and support in the biological brain, the concept imagery in the mathematical mind can be very different from the working of the computational computer. A professional mathematician with mathematical cognitive units may use the computer in a very different way from the student who is meeting new ideas in a computer context.
The article then goes on to explain some of Tall's insights into what/how the brain is learning math in a graphing calculator or computer environment versus in the traditional paper and pencil environment. One way to think of this is in terms of the automation of tasks. A mathematician's brain has automated many tasks, and this has come through a considerable amount of practice. An alternative to this mental automation is, in a number of cases, to learn to use a graphing calculator or computer.
Thus, a math student who is a heavy user of graphing calculators and computers will be developing a type of math maturity that is different than that being developed by a person who does not become proficient in the use of these math tools. Tall does not argue that one type of approach to math education is superior to the other—just that certain aspects of the final results in a student's math-brain will be different. (See also: Moursund, 1986, 1988.)
Automaticity
This section is a work in progress.
Research into how people solve problems and gain in expertise within a particular problem-solving domain have helped us to understand how study and practice lead to increased automaticity and less demands on one's brain. Thus, for example, an expert chess player can recognize and process possible desirable moves in a complex board position much more rapidly and accurately than can an less qualified chess player. This speed of recognition and analysis has come from many thousands of hours of study and practice.
A similar type of learning occurs in math. Through thousands of hours of study and practice, a mathematician's brain automates a large number of problem recognition and possible action tasks. Thus, when faced by a new and challenging math problem, the expert mathematician is able to devote more brain power to the new and challenging parts while automaticity takes care of the familiar parts.
The chess and math examples apply to all areas in which a person can achieve a high level of expertise.
"As a task to be learned is practiced, its performance becomes more and more automatic; as this occurs, it fades from consciousness, the number of brain regions involved in the task becomes smaller." (A Universe Of Consciousness How Matter Becomes Imagination. Edelman & Tononi, 2000, p.51)
That is, through repetition of a task, a brain becomes more efficient at carrying out the task. But, it can take a lot of repetitions plus occasional practice to build and maintain this efficiency. An alternative in some cases is to just turn such repetitive task over to a computer or a calculator. Quoting from David Tall's 1996 article Can All Children Climb the Same Curriculum Ladder?:
This presentation presents evidence that the way the human brain thinks about mathematics requires an ability to use symbols to represent both process and concept. The more successful use symbols in a conceptual way to be able to manipulate them mentally. The less successful attempt to learn how to do the processes but fail to develop techniques for thinking about mathematics through conceiving of the symbols as flexible mathematical objects. Hence the more successful have a system which helps them increase the power of their mathematical thought, but the less successful increasingly learn isolated techniques which do not fit together in a meaningful way and may cause the learner to reach a plateau beyond which learning in a particular context becomes difficult.
The computer is quite different from the biological brain and therefore can be of value by providing an environment that complements human activity. Whilst the brain performs many activities simultaneously and is prone to error, the computer carries out individual algorithms accurately and with great speed. Computer calculations with numbers and manipulation of symbols has some similarities with the notion of procept. Internal computer symbolism is used both to represent data and also to perform routines to manipulate that data. However, there are significant differences. The computer is simply a device which manipulates information in a way specified by a program. It has none of the cognitive richness (or baggage) of the concept image available to the human mind to guide (or confuse) problem-solving activities.
This quote captures some of the idea of students learning by rote (sort of like a computer) and students learning with understanding as well with mental links to related topics that they know something about. This richer learning is a goal in math education. Here is a related quote from Tall:
The development of symbol sense throughout the curriculum therefore faces a number of major reconstructions which cause increasing difficulties to more and more students as they are faced with successive new ideas that require new coping mechanisms. For many it leads to the satisfying immediate short-term needs of passing examinations by rote-learning procedures. The students may therefore satisfy the requirements of the current course and the teacher of the course is seen to be successful. However, if the long-term development of rich cognitive units is not set in motion, short-term success may only lead to increasing cognitive load and potential long-term failure.
Learners have changed as a result of their exposure to technology, says Greenfield, who analyzed more than 50 studies on learning and technology, including research on multi-tasking and the use of computers, the Internet and video games. Her research was published this month in the journal Science.
Reading for pleasure, which has declined among young people in recent decades, enhances thinking and engages the imagination in a way that visual media such as video games and television do not, Greenfield said.
…
Visual intelligence has been rising globally for 50 years, Greenfield said. In 1942, people's visual performance, as measured by a visual intelligence test known as Raven's Progressive Matrices, went steadily down with age and declined substantially from age 25 to 65. By 1992, there was a much less significant age-related disparity in visual intelligence, Greenfield said.
"In a 1992 study, visual IQ stayed almost flat from age 25 to 65," she said.
Once Mischel began analyzing the results, he noticed that low delayers, the children who rang the bell quickly, seemed more likely to have behavioral problems, both in school and at home. They got lower S.A.T. scores. They struggled in stressful situations, often had trouble paying attention, and found it difficult to maintain friendships. The child who could wait fifteen minutes had an S.A.T. score that was, on average, two hundred and ten points higher than that of the kid who could wait only thirty seconds.
New research funded by the Economic and Social Research Council (ESRC) and conducted by Michael Shayer, professor of applied psychology at King's College, University of London, concludes that 11- and 12-year-old children in year 7 are "now on average between two and three years behind where they were 15 years ago", in terms of cognitive and conceptual development.
"It's a staggering result," admits Shayer, whose findings will be published next year in the British Journal of Educational Psychology. "Before the project started, I rather expected to find that children had improved developmentally. This would have been in line with the Flynn effect on intelligence tests, which shows that children's IQ levels improve at such a steady rate that the norm of 100 has to be recalibrated every 15 years or so. But the figures just don't lie. We had a sample of over 10,000 children and the results have been checked, rechecked and peer reviewed."
This book includes an emphasis on thinking about problem solving partly from the point of view of developing a repertoire of smaller problems or problem-solving activities to a high level of automaticity or making use of comptuters as an substitute for part of this learning task.
Author or Authors
Math Maturity, Defined
Mathematicians tend to prefer the concept of math maturity over the idea of math cognitive development. A Google search (10/6/08) of the expression: "math maturity" OR "mathematical maturity" OR "mathematics maturity" produced over 24,000 hits. Wikipedia states:
Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts.
Still quoting from the Wikipedia, other aspects of mathematical maturity include:
the capacity to generalize from a specific example to broad concept
the capacity to handle increasingly abstract ideas
the ability to communicate mathematically by learning standard notation and acceptable style
a significant shift from learning by memorization to learning through understanding
the capacity to separate the key ideas from the less significant
the ability to link a geometrical representation with an analytic representation
the ability to translate verbal problems into mathematical problems
the ability to recognize a valid proof and detect 'sloppy' thinking
the ability to recognize mathematical patterns
the ability to move back and forth between the geometrical (graph) and the analytical (equation)
improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude
Thirty percent of mathematical maturity is fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas. Mathematics, like English, relies on a common understanding of definitions and meanings. But in mathematics definitions and meanings are much more often attached to symbols, not to words, although words are used as well. Furthermore, the definitions are much more precise and unambiguous, and are not nearly as susceptible to modification through usage. You will never see a mathematical discussion without the use of notation!
You can evaluate a math lesson plan or unit of study in terms of how it contributes to students gaining in math maturity.
The general notion of "maturity" in a discipline applies to every discipline—indeed to every job, vocation, or pastime. However, mathematics teachers have been engaged with the notion more often than teachers of other academic disciplines.
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MATH 7: All the expected algebraic topics are covered in this text. Patterns, relations, and functions are presented early in the text and are reviewed and practiced throughout the year. Order of operations are applied to whole numbers, integers, rational numbers, and exponents. Students build on their understanding of variables and expressions and extend them to equations and inequalities. Students also analyze patterns and functions leading to graphing on the coordinate plane.
MATH 8: Similar to Course 2 however the development of algebraic thinking progresses from Course 1 to Course 3, building a solid foundation for students to have confidence and success in Algebra I.
ALGEBRA I: Saxon Algebra 1 covers advanced topics such as arithmetic of and evaluation of expressions involving signed numbers exponents and roots. Students learn properties of the real numbers; absolute value and equations or inequalities involving absolute value; unit conversions; solution of equations in one unknown and solution of simultaneous equations; polynomials and rational expressions; word problems requiring algebra; Pythagorean theorem; functions and functional notation; solution of quadratic equations; and much, much more.
GEOMETRY: Saxon Geometry books teach postulates and theorems and two column proofs. They also teach triangle congruence, surface area and volume, vector addition, slopes and equations of lines. With topics like these, Saxon Geometry books cover all the ground of a traditional high school geometry course, with some additional topics thrown in to connect with real life applications as well as Algebra review.
ALGEBRA II: Saxon Algebra 2 topics covered include: graphical solution to simultaneous equations; roots of quadratic equations, even including complex roots; inequalities and systems of inequalities; logarithms and antilogarithms; exponential equations; basic trigonometric functions; vectors; polar and rectangular coordinate systems, and so much more! There are also many different types of word problems requiring algebra in their solution, and real world applications in areas such as physics and chemistry are discussed.
Saxon Algebra 2 books are rather unique in that they not only cover second year algebra, but also a good deal of geometry, equaling about a semester's work of informal geometry, including proof outlines. There is also treatment of set theory and probability and statistics.
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Properties: Comprehend/Apply The learner will be able to
comprehend and apply the distributive, associative, commutative, inverse, identity, and substitution property to evaluate algebraic expressions.
Expressions: Procedures/Apply The learner will be able to
apply procedures for operating on algebraic expressions, commutative, associative, identity, zero, inverse, distributive, substitution, multiplication over addition.
Functions: Linear/Equations/Inequalities The learner will be able to
create equations and/or inequalities that are based on linear functions, apply many different methods of solution, and/or study the solutions in the context of the situation.
Figures: Two-/Three-Dimensional Objects The learner will be able to
precisely explain, classify, and comprehend relationships among types of two- and three-dimensional objects by applying their defining properties.
Math Concepts: Identify The learner will be able to
recognize concrete and/or symbolic illustrations of vertical, supplementary, complementary, and straight angles, parallel and perpendicular lines, transversals, and/or special quadrilaterals, and apply them to obtain solutions to problems.
Mathematical Reasoning: Explain The learner will be able to
apply many different methods to describe mathematical reasoning such as words, numbers, symbols, graphical forms, charts, tables, diagrams, and/or models.
Area/Volume/Length: Differences The learner will be able to
identify the differences and relationships between perimeter, area, and volume (capacity) measurement in the metric and U.S. Customary measurement systems.
Surface Area/Volume: Compute/Solve The learner will be able to
compute the surface area and volume of pyramids, cylinders, cones, and spheres and obtain solutions to problems involving volume and surface area.
Units: Choose/Metric/Customary The learner will be able to
choose suitable customary and metric measurement units for length (include perimeter and circumference), area, capacity, volume, weight, mass, time, and temperature.
Magnitude: Illustrate/Understanding The learner will be able to
illustrate an understanding of magnitudes and relative magnitudes of real numbers (integers, fractions, decimals) using scientific notation and exponential numbers.
Strategies: Daily life/Apply The learner will be able to
apply various methods, including common mathematical formulas, to obtain problem solutions of routine and non-routine problems drawn from everyday life.
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Haverford TrigonometryGraph theory deals with the study of graphs and networks and involves terms such as edges and vertices. This is often considered a very specific branch of combinatorics. Lastly, probability in discrete math deals with events that occur in discrete sample spaces
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Synopses & Reviews
Publisher Comments:
Based on the premise that in order to write proofs, one needs to read finished proofs as well as study both their logic and grammar, Revolutions in Geometry depicts how to write basic proofs in various fields of geometry. This accessible text for junior and senior undergraduates explains the general development of geometry throughout time, discusses the involvement of its major contributors, and places the proofs into the context of geometry's history to illustrate how crucial proof writing is to the job of a mathematician
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Windows Software
equationsThis bilingual problem-solving mathematics software allows you to work through 19292 trigonometric equations with guided solutions, and encourages to learn through in-depth understanding of each solution step and repetition rather than through rote memorization. The software includes all trigonometr...
Qds Equations - equation editor is a set of visual components for Delphi that allow to enter and display formulas of any complexity, from simple Greek symbols to matrixes and complex integral expressions. You can use the equation editor in your projects written in the Delphi environment, for example... ...
This bilingual problem-solving mathematics software allows you to work through 84102 trigonometric problems with guided solutions, and encourages to learn through in-depth understanding of each solution step and repetition rather than through rote memorization. The software offers tasks on simplific ...
MatBasic is a calculating, programming and debugging environment using special high-level programming language designed for solving mathematical problems. MatBasic programming language allows execution of difficult mathematical calculations, involving an exhaustive set of tools for the purpose of cr ...
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Math Refresher for Trades & Technology
Course Overview
Improve your math skills. This course will provide a solid foundation before starting a trades or technology program. It will be very helpful for those enrolled in many pre-apprenticeship or apprenticeship programs, but is suitable for anyone who wants a refresher whether you are a recent high school graduate or a mature student.
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Decent instructor, but not very good in the English dept. Pretty fair grading standards, and tests were reasonable. He just has a hard time explaining things sometimes because his communication skills are kind of lacking; also, he's obviously brilliant, and doesn't always understand why some students don't get everything that seems clear to him.
Course is designed as a survey of the kinds of math included on collegiate mathematics contests, specifically the Putnam exam. As such, the material jumps from topic to topic quickly, and is taught by up to five professors in any given semester. Pretty disjointed, but interesting to people who like "neat" math problems and want to take the Putnam, but probably not of any use to anyone else.
He may be difficult to understand at times -- not really because of the language, but because he can't explain things too well. However, he is a very friendly and nice teacher, and that always is a good thing.
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Our Price$38.24Reg.$44.99Learning Wrap Ups Pre Algebra Intro Kit
This kit contains 600 problems and answers in all, that will help students master the basic concepts of pre-algebra. Concepts that are covered include: addition, subtraction, multiplication, and division of positive & negative... more
Our Price$66.05Reg.$77.70Saxon Algebra 1 2 Home School Kit
Algebra 1/2 represents a culminatin of prealgebra mathematics, covering all topics normally taught in prealgebra, as well as additional topics from geometry and discrete mathematics. This program is recommended for seventh... more
Our Price$7.61Reg.$8.95Pre Algebra Flipper8.46Reg.$9.95Spectrum Algebra 6-8
Excellent Tool For Standardized Test Preparation
Spectrum Algebra helps students from sixth through eighth grade improve and strengthen their math skills in areas such as factors and fractions; equations and inequalities; functions and graphing; rational numbers; and... more
Our Price$79.16Reg.$87.95Horizons Pre Algebra Complete Set
Give your child a smooth transition into advanced math with the Horizons Pre-Algebra Set! This popular math curriculum builds on basic math operations with hands-on lessons in basic algebra, trigonometry, geometry, and real-life... more
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Data Points: Visualization That Means Something
by Nathan Yau Publisher Comments
Reveal the story your data has to tell To create effective data visualizations, you must be part statistician, part designer, and part storyteller. In his bestselling book Visualize This, Nathan Yau introduced you to the tools and programming techniques... (read more)
Analysis of Longitudinal Data
by Peter Diggle Publisher Comments
The first edition of Analysis for Longitudinal Data has become a classic. Describing the statistical models and methods for the analysis of longitudinal data, it covers both the underlying statistical theory of each method, and its application to a range... (read more)
Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces
by N. J. Hitchin Publisher Comments
This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are... (read more)
Brain Teasers, Puzzles & Mathematical Diversions
by Erwin Brecher Publisher Comments
Hundreds of puzzles to give any puzzler's brain a great workout For those who enjoy putting their mental agility to the test, this book offers a superb collection of many types of puzzle, all crammed into this one volume. There are riddles, lateral... (read more)
A History of Mathematics: From Mesopotamia to Modernity
by Luke Hodgkin Publisher Comments
A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians.... (read more)
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Piecewise functions Teacher Resources
Title
Resource Type
Views
Grade
Rating
Students explore the concept of piecewise functions. In this piecewise functions lesson, students graph piecewise functions by hand and find the domain and range. Students make tables of values given a piecewise function. Students write piecewise functions given a graph.
A hands-on lesson using the TI-CBR Motion Detector to provide information to graph and analyze. The class uses this information to calculate the slope of motion graphs and differentiate scalar and vector quantities. There is a real-world activity of a Roof Manufacturer's Test in regards to the pitch of roofs, as well as several other real-world scenarios.
Learners explore the concept of piecewise functions. In this piecewise functions lesson, students discuss how to make a piecewise function continuous and differentiable. Learners use their Ti-89 to find the limit of the function as it approaches a given x value. Students find the derivative of piecewise functions.
Young scholars explore the concept of piecewise functions. In this piecewise functions lesson, students write functions to represent the piecewise function graphs on their Ti-Nspire calculator. Young scholars determine the formula given the piecewise function graph.
Graph piecewise functions as your learners work to identify the different values that will make a piecewise function a true statement. They identify function notations and graph basic polynomial functions. This lesson plan includes a series of critical thinking questions and vocabulary.
Students explore piecewise functions. In this Algebra II/Pre-calculus lesson, students write formulas for piecewise functions and check their work on the calculator. The lesson assumes that students have seen piecewise functions prior to this activity.
Calculus students find the limit of piecewise functions at a value. They find the limit of piecewise functions as x approaches a given value. They find the limit of linear, quadratic, exponential, and trigonometric piecewise functions.
This pre-calculus worksheet is short, yet challenging. High schoolers calculate the limit of piecewise functions, rational functions, and graphs as x approaches a number from the positive or negative side. There are four questions.
Learners explore the concept of piecewise functions. In this piecewise functions instructional activity, students find the derivatives of piecewise functions. Learners determine points of discontinuity and jumps in the graph using their Ti-89 calculator.
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Mathematics for Elementary Teachers, Third Edition offers an inquiry-based approach, which helps readers reach a deeper understanding of mathematics. Sybilla Beckmann, known for her contributions in math education, writes a text that encourages future teachers to find answers through exploration and group work. Fully integrated activities are found in her accompanying Activities Manual, which comes with every new copy of this text. As a result, readers engage, explore, discuss, and ultimately reach a true understanding of mathematics.
Customer Reviews:
excellent book for learning to think about elementary math
By mathwonk - August 29, 2010
The recent review of books and programs in mathematics teacher education in America by the National Council on Teacher Quality rated this book the best one in the country, and rated the mathematics education program at UGA where this book is used and was developed, as the ONLY "exemplary" program in the entire nation. Obviously not everyone finds it easy to read, but it does serve its purpose of helping, and requiring, the student to understand the concepts behind the mathematics. It is not easy to teach from this book, as I have learned by experience. The reason was not the fault of the book, but rather the extreme difficulty in getting "test oriented" students to stop looking at mathematics as just a list of formulas and procedures, and to begin trying to grasp the ideas which underly them. Understanding is harder than calculating, and having read it, I can easily understand why this book is professionally considered the best one in the country for making that transition in... read more
Satisfied Student
By Katie Driver "Katie" - October 12, 2011
This book, and the activity book that comes with it, was required for an education class I'm taking, and I found it very interesting. I plan on keeping it to use in the future when I become a teacher; the activity book included has a large variety of activities that can be used to help students learn. I highly recommend this book if you are interested in teaching any form of math in the future.
Great book for my course
By Monica Caropreso - January 9, 2013
This manual was invaluable in my course in college. Its activities really helped nail down the concepts my professor taught.
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To succeed in the use or study of advanced math (algebra and above), the student must acquire the disciplines of care and rigor, in reading math as well as in writing it. The very precision that makes math so useful leads to...
Often the biggest stumbling block for students beginning algebra and other advanced math subjects is the unfamiliar nomenclature, including specialized definitions of what appear to be familiar terms. The advanced math...
Most failures in advanced math studies result from inadequate study discipline on the part of the student. Such students have a harder and harder time with math as they progress, due to insufficient mastery of the earlier...
This course presents the basics of writing mathematical expressions using algebraic symbols, and of solving equations, all from the viewpoint that math is a language used to express, develop and communicate ideas. The point...
Following on Algebra 1 Part A, this course covers how mathematical equations with two variables (x and y) can be depicted in graphical format, and how the solution of two equations can be found by where their lines cross on...
The theme of Algebra 2 is "using mathematics to predict."
Polynomials are equations with more than one term, quadratic equations have a term with a squared unknown. The student learns how such equations may be added,...
A student who has completed a second year of algebra study is ready to learn more practical uses for the subject.
This course is the final course in the Heron algebra series. It gives the student a good feel for the use of...
This manual is made available as an aid to supervisors of the Heron Geometry courses (Plane Geometry and Solid Geometry).
The manual provides fully worked out solutions to exercises and selected course step...
The course covers linear and quadratic functions and includes data sheets introducing the use of a graphing calculator to solve systems of equations by the use of matrices. The course also introduces the use of imaginary and...
The course covers exponential, logarithmic, rational and irrational functions, familiarizing the student with rest of "the five algebraic functions" and their applications.
The course is based on the Algebra and...
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About the Book
Since everything assembled consists of either straight lines, curved lines, or a combination of both, the ability to calculate circles and right triangles is essential for anyone who works in a building trade. This simple and straightforward book explains the basic math used in construction, manufacturing, and design. Starting with fractions and decimals and moving to mitered turns and arcs, these principles are presented with detailed illustrations, practical applications, and in larger print for easy reading. The result is increased efficiency, productivity, and confidence in one's work from initial design to final product.
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12th class functions maths notes
Lecture Notes on Actuarial Mathematics
§0. Introduction. The objective of these notes is to present the basic aspects of the theory of .... Finding the nominal interest rate i which makes these two functions equal is now an easy ...... What is the loan balance at the end of the 12th month?
Lecture Notes on the Mathematics of Finance
The central theme of these notes is embodied in the question, "What is the value today ..... continuous variable and show that the right hand side is a decreasing function of m for fixed ...... What is the loan balance at the end of the 12th month?
http// This is a set of lecture notes for introductory courses in mathematical logic offered at ...... 2. a function l T → N+,. 3. a binary ...
Highlights From TIMMS
by F Grade
Class Field Theory
These notes contain an exposition of abelian class field theory ..... one of the most beautiful ideas in mathematics for generating abelian extensions of number ... 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to ... Studied the Riemann zeta function, and made the Riemann hy- ...
Mathematics of Program Construction, volume 2297 of Lecture Notes in Computer ... Introduction to Algebraic and Coalgebraic Methods in the Mathematics of ... Proceedings of the 12th International Conference on Theorem Proving in Higher ...
HISTORY OF CLASS FIELD THEORY - Department of Mathematics ...
An important case of Kronecker's work uses the j-function if K is imaginary ...... This made class field theory much more accessible, but note the year it was ...... [17] N. Schappacher, On the History of Hilbert's 12th Problem A Comedy of Errors, ...
Educational Aspects of Mathematics Disabilities Susan Peterson ...
The mean math scores of 12th grade students in both studies (Cawley. & Miller, 1989 ... arithmetic and problem solving using discovery methods (Carnine, 1992). During the ... in the eighth grade. That figure was up from the 13% noted in 1981 ...
tions, and functions that make up much of mathematics and its applications. Algebra, both as a ... Teaching and Learning Algebra in Kindergarten Through 12th Grade 45 ..... As Kieran (1992) notes, "The research commu- nity knows very little ...
Discretization Techniques A recent survey
Department of Mathematics, University of Patras, Greece sotos@math.upatras.gr, dkanellop@teipat.gr. Abstract. A discretization .... In contrast, discretization methods that use class labels for ..... Lecture Notes in Computer Science 2724, 1818-1831 ... In Proceedings of the 12th international conference on machine learning.and
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Learn techniques to be successful in math courses! Learn what online resources are available, learn about problem-solving techniques, the ins and outs of graphing and how to effectively use the graphing calculator.
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The high school math curriculum helps students to prepare for college. Every student is required to accumulate 3 math credits in order to graduate. Every student must pass the Integrated Algebra Regents exam in order to graduate. If a student wishes to pursue an Advanced Academic diploma with designation, they must past the Geometry and Algebra II/Trig Regents exams also.
Choose the appropriate link to view teachers, courses, and links to other sites that will help students and parents succeed.
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Personal tools
Introduction to Algebra: Content and Instruction (BOS)
Click on the register button to sign up for this event.
Click on the print icon to see a printer-friendly view of this event.
Overview
This workshop will focus on the algebraic thinking needed by ABE/ASE students. Our students need a strong foundation in algebraic thinking and practices to keep doors to the future open, but research tells us that students' skills are often weak. You can help. No matter how long it's been since your own last "algebra encounter," no matter whether you see it as a helpful tool or a monstrous challenge, you will enjoy the camaraderie of learning in this workshop. Join with colleagues in exploring the meaning of algebra and paths to better algebra instruction that foster communication, problem solving, and reasoning--essential skills to prepare ABE/ASE students for good jobs and further education. In this workshop, we will investigate general math teaching principles, as well as algebra-specific ideas. Together we will solve algebra problems suitable for lessons. Classroom materials are provided.
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The following Internet sites contain mathematics
contests and competitions, mathematics problems, archives of problems, or
information about problem resources. Problems of all levels from early childhood
mathematics through graduate level mathematics are listed. The intention is to
identify those unique resources made available at each site.
MATH Challenge (Michigan
Autumn Take Home Challenge)
The Michigan Autumn Take Home Challenge is a team-oriented math competition
for undergraduates. Teams of 2 or 3 students take a 3-hour exam consisting
of 10 interesting problems dealing with topics and concepts found in the undergraduate
mathematics curriculum. Each team takes the exam on their home campus under
the supervision of a faculty advisor.
American Mathematics Competitions
The American Mathematics Competitions (AMC) seek to increase interest in mathematics
and to develop problem solving ability through a series of friendly mathematics
contests for junior (grades 8 and below) and senior high school students (grades
9 through 12).
American Regions Mathematics League
The American Regions Mathematics League (ARML) is an annual national mathematics
competition. High school students form teams of 15 to represent their city,
state, county or school and compete against the best in the from the United
States and Canada.
The Bay Area Mathematical
Olympiad and Mathematical Circles
The Bay Area Mathematical Olympiad (BAMO) is a contest for high school students
sponsored jointly by the Mathematical Sciences Research Institute (MSRI),
the American Institute of Mathematics (AIM), the University of California
at Berkeley (UCB), and the University of San Francisco (USF).
Colorado Mathematical
Olympiad
The COLORADO MATHEMATICAL OLYMPIAD is the largest essay-type mathematical
competition in the United States, with 600 to 1,000 participants competing
annually for prizes
Descartes
Mathematics Contest
The Contest is designed primarily for outstanding students in their graduating
year of high school, but should also be considered for other exceptional students.
The aim of the contest is to provide these students with an opportunity to
test their mathematical ability.
Great Plains Math League
The Great Plains Math League will be sponsoring a series of high school math
contests to determine the individual and team state champions for Missouri,
Iowa and Kansas in several math events.
Harvard-MIT Math Tournament
The Harvard-MIT Math Tournament is an annual math tournament for high school
students, held in alternating years at MIT and at Harvard. It is run by
MIT and Harvard students who participated in math contests in high school,
so try to incorporate what we liked best about math contests when we were
in high school.
High
School Mathematical Contest in Modeling
The purpose of The High School Mathematical Contest in Modeling is to offer
students students the opportunity to compete in a team setting using mathematics
to present solutions to real-world modeling problems.
International Mathematical Olympiad
The aims of the International Mathematical Olympiad are: to discover, encourage
and challenge mathematically gifted young people in all countries; to foster
friendly relations between mathematicians of all countries; to create an opportunity
for the exchange of information on school mathematics syllabi and practice
in mathematics education throughout the world. To achieve these aims, an international
contest for secondary students in solving mathematical problems is held each
year.
Related sites:
International Mathematics
Tournament of Towns
The International Mathematics Tournament of Towns is a mathematics problem
solving competition in which towns throughout the world can participate on
an equal basis. The Tournament is open to all high school students, with the
highest age of students being about 17 years old.
Mathematical Challenge
Mathematical Challenge is a problem solving competition for individual pupils
in Scottish secondary schools and upper primary schools (mainly P7). The aim
is to promote mathematics as a source of interest and pleasurable achievement.
Mathematics Olympiad Learning Centre
A commercial site operated by Arkadii Slinko who has had experiences with
the Math Olympiads in both Russia and New Zealand. A lot of free material
is available from the site including Problems of the IMO 2001 (with official
solutions), NZ Math Olympiads, tutorials and articles. In addition, a Monthly
Internet Olympiad is supported.
Rice University
Mathematics Tournament
The Rice University Mathematics Tournament is an annual event for high school
students and
teams from schools in the Texas/Louisiana area. Organized by Rice undergraduate
students, the Rice Mathematics Tournament has been held almost every year
at the Rice University campus in Houston since 1981. The contest includes
individual written tests as well as group events.
St.
Cloud State University Math Contest
For the past 34 years, the St. Cloud State departments of mathematics, statistics,
and computer science have held a mathematics contest for students in grades
7-12.
Stanford Math Tournament
This contest brings together students from the greater Bay Area for a day
of challenging
problems and mathematical fellowship. The tournament is composed of four subject
area tests (algebra, geometry, advanced topics, and calculus), a general test,
and a team test.
Southern Illinois University
Mathematics Field Day Each Spring, the Mathematics Department sponsors a mathematics contest,
called Math Field Day, for regional high school students. Schools from the
southern part of Illinois, ranging from Edwardsville to Fairfield to Cairo,
as well as a few schools from Missouri, Kentucky, and Tennessee participate.
University of
Canberra Maths Day
The best Year 12 mathematics students from 37
schools from the ACT and southern NSW tested their wits in
a variety of team events.
USA Mathematical
Talent Search (USAMTS)
The purpose of the USAMTS is to encourage and assist the development of problem
solving skills of talented high school students. While most other competitions
have stringent time limits, the USAMTS allows for more reflection on the part
of the participants. We wish to foster not only insight, ingenuity, quick
thinking, and creativity, but also writing skills and the virtue of perseverance.
Math Forum - Problems of the
Week
The Math Forum's Problems of the Week (POWs) are designed to provide creative,
non-routine challenges for students in grades three through twelve. Problem-solving
and mathematical communication are key elements of every problem.
Wisconsin Mathematics,
Engineering and Science Talent Search
Each school year, the Talent Search creates five sets of five problems each
and distributes them to high school and middle school students in the state
of Wisconsin and throughout the world. These problems are unusual, challenging,
and we hope, enjoyable. They are not easy, but their solutions do not require
advanced mathematical knowledge--just talent in problem solving.
Geombinatorics
A journal of OPEN PROBLEMS of combinatorial and discrete geometry and related
areas.
MathPro Online - Index
to Journal and Contest Problems
Thousands of challenging problems are published each year in contests and
problem columns around the globe. MathPro Online provides an easy way to search
many of these problems. References to published solutions are included, but
solutions themselves are not included. The database contains 20,946 math problems
from 38 journals and 21 contests.
KöMaL -
Mathematical and Physical Journal for Secondary Schools
It was more than a hundred years ago that Dániel Arany, a high school
teacher from the city of Gyõr, decided to found a mathematical journal
for high school students. His goal was "to give a wealth of examples
to students and teachers". The journal's first edition appeared on January
1, 1894. From that time several generations of mathematicians and scientists
developed their problem-solving skills through KöMaL.
Mathematics and
Informatics Quarterly
The M&IQ is an international journal which publishes articles, notes, problems
and solutions in school mathematics and informatics and is devoted to teachers
and students, who are interested in mathematics and informatics clubs, olympiads
and competitions.
Other Related Sites
World Federation
of National Mathematics Competitions The World Federation of National Mathematics Competitions is an
organisation of national mathematics competitions affiliated as a Special
Interest Group of the International Commission for Mathematical Instruction
(ICMI).
Acknowledgment: Most of the descriptions
above were taken from the listed sites.
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Elmhurst, IL CalculusDefinitions, Postulates, Theorems, and Proofs meets the world of polygons and circles. By now, you know you can figure out answers, but do you know *why* those answers are right? Can you break it down and provide evidence at each step?
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Description
In Precalculus, the authors encourage graphical, numerical, and algebraic modeling of functions as well as a focus on problem solving, conceptual understanding, and facility with technology. They have created a book that is designed for instructors and written for students making this the most effective precalculus text available today.
Table of Contents
Contents:
P. Prerequisites
1. Functions and Graphs
2. Polynomial, Power, and Rational Functions
3. Exponential, Logistic, and Logarithmic Functions
4. Trigonometric Functions
5. Analytic Trigonometry
6. Applications of Trigonometry
7. Systems and Matrices
8. Analytic Geometry in Two and Three Dimensions
9. Discrete Mathematics
10. An Introduction to Calculus: Limits, Derivatives, and Integrals
Appendix A: Algebra Review
Appendix B: Key Formulas
Appendix C: Logic
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any math good books for physics olympiad
any math good books for physics olympiad
I'm now preparing for IPHO, I would like to know some good calculus reference books for the math it requires. Do I need to know multivariable calculus and differential equations? If yes, what topics are required?
And can some1 recommend some solved problems book/websites that have many solved problems for multivariable calculus/differential equations?
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complete, interactive, objective-based approach, "Intermediate Algebra: An Applied Approach," is a best-seller in this market. The Seventh Edition provides mathematically sound and comprehensive coverage of the topics considered essential in an intermediate algebra course. An Instructor''s Annotated Edition features a comprehensive selection of instructor support materials. The Aufmann Interactive Method is incorporated throughout the text, ensuring that students interact with and master the concepts as they are presented. This approac... MOREh is especially important in the context of rapidly growing distance-learning and self-paced laboratory situations."Study Tips" margin notes provide point-of-use advice and refer students back to the "AIM for Success" preface for support where appropriate."Integrating Technology" margin notes provide suggestions for using a calculator in certain situations. For added support and quick reference, a scientific calculator screen is displayed on the inside back cover of the text."Aufmann Interactive Method (AIM)" Every section objective contains one or more sets of matched-pair examples that encourage students to interact with the text. The first example in each set is completely worked out; the second example, called ''You Try It, '' is for the student to work. By solving the You Try It, students practice concepts as they are presented in the text. Complete worked-out solutions to these examples in an appendix enable students to check their solutions and obtain immediate reinforcement of the concept. While similar texts offer only final answers to examples, the Aufmann texts'' complete solutions help students identify their mistakes and preventfrustration."Integrated learning system organized by objectives." Each chapter begins with a list of learning objectives that form the framework for a complete learning system. The objectives are woven throughout the text (in Exercises, Chapter Tests, and Cumulative Reviews) and also connect the text with the print and multimedia ancillaries. This results in a seamless, easy-to-navigate learning system."AIM for Success" Student Preface explains what is required of a student to be successful and demonstrates how the features in the text foster student success. "AIM for Success" can be used as a lesson on the first day of class or as a project for students to complete. The Instructor''s Resource Manual offers suggestions for teaching this lesson. "Study Tip" margin notes throughout the text also refer students back to the Student Preface for advice."Prep Tests" at the beginning of each chapter help students prepare for the upcoming material by testing them on prerequisite material learned in preceding chapters. The answers to these questions can be found in the Answer Appendix, along with a reference (except for chapter 1) to the objective from which the question was taken, which encourages students who miss a question to review the objective."Extensive use of applications" that use real source data shows students the value of mathematics as a real-life tool."Focus on Problem Solving" section at the end of each chapter introduces students to various problem-solving strategies. Students are encouraged to write their own strategies and draw diagrams in order to find solutions. These strategies are integrated throughout the text. Several open-ended problems are included, resulting in morethan one right answer and strengthening problem-solving skills."Unique Verbal/Mathematical connection" is achieved by simultaneously introducing a verbal phrase with a mathematical operation. Exercises following the presentation of a new operation require that students make a connection between a phrase and a mathematical process."Projects and Group Activities" at the end of each chapter offer ideas for cooperative learning. Ideal as extra-credit assignments, these projects cover various aspects of mathematics, including the use of calculators, collecting data from the Internet
Chapters 2–6 are followed by Cumulative Review Exercises
Review of Real Numbers
Introduction to Real Numbers
Operations on Rational Numbers
Variable Expressions
Verbal Expressions and Variable Expressions
Focus on Problem Solving: Polya's Four-Step Process Projects and Group Activities: Water Displacement
Focus on Problem Solving: Another Look at Polya's Four-Step Process Projects and Group Activities: Solving Radical Equations with a Graphing Calculator, The Golden Rectangle
Quadratic Equations
Solving Quadratic Equations by Factoring or by Taking Square Roots
Solving Quadratic Equations by Completing the Square
Solving Quadratic Equations by Using the Quadratic Formula
Solving Equations that are Reducible to Quadratic Equations
Quadratic Inequalities and Rational Inequalities
Applications of Quadratic Equations
Focus on Problem Solving: Inductive and Deductive Reasoning Projects and Group Activities: Using a Graphing Calculator to Solve a Quadratic Equation
Functions and Relations
Properties of Quadratic Functions
Graphs of Functions
Algebra of Functions
One-to-One and Inverse Functions
Focus on Problem Solving: Proof in Mathematics Projects and Group Activities: Finding the Maximum or Minimum of a Function Using a Graphing Calculator, Business Applications of Maximum and Minimum Values of Quadratic Functions
Exponential and Logarithmic Functions
Exponential Functions
Introduction to Logarithms
Graphs of Logarithmic Functions
Solving Exponential and Logarithmic Equations
Applications of Exponential and Logarithmic Functions
Focus on Problem Solving: Proof by Contradiction Projects and Group Activities: Solving Exponential and Logarithmic Equations Using a Graphing Calculator, Credit Reports and FICOreg; Scores
Conic Sections
The Parabola
The Circle
The Ellipse and the Hyperbola
Solving Nonlinear Systems of Equations
Quadratic Inequalities and Systems of Inequalities
Focus on Problem Solving: Using a Variety of Problem-Solving Techniques Projects and Group Activities: The Eccentricity and Foci of an Ellipse, Graphing Conic Sections Using a Graphing Calculator
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Mathematics: Title III Grant
Title III Grant: Mathematics Foundation and STEM Success
The Title III Mathematics Foundation and STEM Success grant funds a five-year project designed to improve student success in mathematics and establish the quantitative fluency essential for success in science, technology, and engineering courses. The project began in October 2008 and runs through September 2013 and directly affects all faculty members in the STEM disciplines: science, technology, engineering, and math.
Central to the project is the mapping and re-design of the core mathematics sequence. From developmental to advanced math, course competencies are mapped to identify gaps, inconsistencies, and unnecessary redundancies. The result is curriculum realignment and courses are being re-designed to incorporate the best research-based practices for student success.
In conjunction with pilot courses, the project is working to improve on-campus learning support services for mathematics including the Mathematic Department's "Math Assistance Center" (MAC) and Freshman Math Program's "Algebra Alcove," along with the class, Independent Studies in Mathematics (ISM), which offers individualized instruction to help students be successful in their math courses.
Because math competency plays a crucial role in student success in all STEM programs, the math faculty and Title III staff collaborate with faculty in other STEM disciplines to integrate key mathematical concepts into all STEM courses. The second year of the project (2009-2010) focused on Biology courses, the third year (2010 – 2011) on Chemistry courses, the fourth year (2011-2012) focuses on Geosciences, and the fifth year (2012 – 2013) on Engineering and Physics.
If you have questions about the Mathematics Foundation and STEM Success, please contact Project Director Rolly Constable: constable_r@fortlewis.edu; 970-247-7234
Independent Studies in Mathematics
Independent Studies in Mathematics (ISM) are classes designed to help Fort Lewis students develop essential math skills through individualized instruction so they are prepared for success in their College Math Courses. Whether a student is having difficulty in a math class, failing a course, or can't enroll in a needed math class, they can enroll in an ISM course and--with a study-plan based on their individual needs--develop math skills in Arithmetic, Geometry, Algebra, Trigonometry, or Pre-Calculus.
ISM is variable credit course--one, two, or three credits, and because it's a late–registration course, students can enroll during the semester--whether or not they are currently enrolled in a Math Class-- and get individual help to succeed in their current or future Math classes.
ISM has limited enrollment and to enroll students will need a referral from their advisor or math instructor.
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A complete Algebra curriculum by the end of eighth grade! This exciting program, developed in cooperation with Education Development Center, Inc., makes mathematics accessible to more of your middle-school students. They will spend less t
Math Skills Maintenance Workbook, Course 2
Editorial review
In order for their skills to remain fresh, students need opportunities to practice the math skills that they have learned in previous courses. Math Skills Maintenance contains pages of practices for various basic math skills. Each page is
Quick Review Math Handbook: Hot Words, Hot Topics provides students and parents with a comprehensive reference of important mathematical terms and concepts to help them build their mathematics literacy. Includes practice problems
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MATH 101 - Calculus
Course Code: MATH 101
Credits: 3
Calendar Description: This course is an introduction to the concepts of differential calculus. Topics include the calculation of limits and the differentiation of functions including trigonometric, inverse trigonometric, exponential, logarithmic functions, chain rule, and implicit differentiation. Further topics include Rolle's theorem, the mean value theorem, differentials, antiderivatives, partial differentiation, as well as the application of differentiation to related rates, maxima and minima, curve sketching, Newton's method. Emphasis on applications is provided for all topics.
Date First Offered: 2007-09-04
Hours: Total Hours: 60
Lecture Hours: 45
Laboratory Hours: 15
Total Weeks: 15
This course is offered online: No
Pre-Requisites: One of Math 100, Math 105, Math 110, "C+" or higher in Principles of Mathematics 12, "C+" or higher in Pre-calculus 12, or C+ or higher in Math 050; alternatively, completion of the Calculus Readiness Assessment.
Learning Outcomes: 1. Review some topics in algebra and analytic geometry such as functions and their graphs.
2. Study of limits: how they arise as slopes of tangents and rates of change; properties of limits; continuity.
3. Study of differentiation: how to calculate derivatives and how to use them to solve problems involving rates of changes
4. Statement and applications of the Mean Value Theorem: How to find maximum and minimum values, points of inflection, asymptotes,
and how to use the above information to sketch curves. How to solve maximum and minimum values.
5. Use the L'Hospital's Rule to evaluate difficult limits.
6. A brief introduction to integral calculus: motivation and definition of definite integrals, use the Fundamental Theorem to evaluate definite integrals, use the Substitution Rule to evaluate indefinite/definite integrals. Use integrals to solve some area problems.
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This ebook provided by RISPs constitutes a collection of forty open-ended investigative activities for use in the A Level pure mathematics classroom. The book consists of two parts: the first part of the ebook lists all activities indexed by topic together with the teachers' notes. The second part advises on how to make the…
The first of two RISP activities, Venn Diagrams explores lines in co-odinate geometry and quadratic functions although the idea can be extended to cover other topics. Students are given a Venn diagram with the sets defined. Students are required to find pairs of lines which fit into each of the eight regions. The task is then repeated…
Odd One Out, provided by RISPs, helps students focus on mathematical properties to determine which of three numbers or expressions is the odd one out.
For example, given the triplet 2, 3, 9:
2 could be the odd one out because it is even and the others are odd.
3 could be the odd one out because it is the only triangle number.
9…
The first of two RISP activities, Modelling the Spread of a Disease requires students to carry out a simulation of a disease spreading. Students carry out an experiment then pool their results to estimate the proportion of the population that will have caught the disease. The experiment is then modelled mathematically using differential…
This RISP activity is in the form of a puzzle in which students are given a general parametric equation with missing coefficients. A number of clues are given such as a point through which the curve passes. Students have to use their understanding of parametric equations to find the missing numbers. There are a variety of different…
The first of two RISP activities, Radians and Degrees, students are set the task of finding an angle whose sine value is the same whether measured in radians or degrees. Solving the problem leads to further discussion about the relationship between radians and degrees and general trigonometric equations.
The second, Generating the…
The first of four RISP activities, Exploring Pascal's Triangle, explores Pascal's triangle, combinations and binomial coefficients. Doing and Undoing the Binomial Theorem and Extending the Binomial Theorem use different approaches to investigate the binomial theorem using negative and fractional indices. Advanced Arithmagons…
Two Repeats covers the revision of algebraic topics including changing the subject of a formula, graphical solution of equations, solving simultaneous and quadratic equations and manipulating surds.
Given a simple starting premise, students have to solve the puzzle by solving a number of different kinds of equations, working logically……
Two Special Cubes is a RISP activity designed to introduce the idea of implicit differentiation. Students are presented with two cubes of length x and y and are told that volume, the surface area and the edge length form an arithmetic progression. Students are asked to find the maximum value for y and hence the corresponding value…
This RISP activity Polynomial Equations with Unit Coefficients sets students the task of finding the roots of polynomials with a large number of terms. Students are required to use a graph plotter to compare the graphs of several polynomials looking for common points and differences. The task leads to a numeriacal method, iteration,…
The first of two RISP activities, Periodic Functions, asks students to write down as many periodic functions as they can. The activity progresses to look at what happens to the period when graphs with different periods are combined. Topics covered are periodic functions, odd functions, even functions, composite functions and transformation…
RISP activity Building Log Equations requires students to form equations given a set of cards and to determine, with examples, whether the equation is always, sometimes or never true and to attempt to say why. Students must include at least one log card in their equation. Students need to be familiar with logs in different number…
This RISP activity gives students four properties - one side is 3cm, one angle is 90 degrees, one side is 4cm and one angle is thirty degrees. Students are required to find as many triangles as they can which contain any three of these four properties. Once the triangles have been found, students are asked to find the area and perimeter…
Two RISP activities designed for students to explore or consolidate ideas about integration.
Introducing e requires students to use a graphing package to explore a variety of functions of the form y equals x to the power of n and attempt to find the value for k for which the area under the graph between 0 and k is exactly one.……
Sequence Tiles requires students to define a position to term rule for a sequence and is extended to iterative sequences, using the set of cards given. Students have to decide the nature of their sequence: convergent and divergent increasing, decreasing, oscillating or periodic.
In Geoarithmetic sequences students are required…
This RISP activity is ideal for introducing, consolidating or revising the idea of proof using a mathematical argument and appropriate use of logiocal deduction.
Students are asked to choose two triangular numbers and find when the difference is a prime number. Students should then be encouraged to attempt to prove their conjecture.…
This RISP activity from can be used when either consolidating or revising ideas of curve-sketching and indices. The numbers phi, e and pi are used in this investigation where students are asked to estimate the size numbers generated when raising these numbers to different powers. It is suggested that a graphing package would prove…
This RISP, Almost Identical, is an activity designed to consolidate work on hyperbolics, exponentials, percentage error and curve sketching. Students are told that the shape of the curve formed by a chain suspended from two points is called a catenary and are asked to attempt to fit a parabola to the curve as best they can, then…
Five RISP starters revise ideas of polynomials and curve-sketching, cover expanding brackets, solving equations graphically, and knowing how to sketch the graphs of curves.
Gold and Silver Cuboid requires the students to use a graph plotter to explore the effects of changing coefficients of a cubic equation. The investigation…
The first of three RISP activities exporing polynomials, The Gold and Silver Cuboid requires students to find a connection between the volume of a cuboid, the surface area and the edge length. The second part of the activity asks students to find the maximum and minimum volume of a cuboid gives certain constraints.
Venn Diagrams…
Seven RISP activities covering a range of topics, each one having some activity which explores coordinate geometry.
Circle Property: Students generate two coordinates. The coordinates form the end points of the diameter of a circle. Students have to find the equation of the circle formed, compare their results with colleagues…
Three RISP activities designed to introduce or consolidate basic algebraic skills.
Brackets Out, Brackets In: students are asked to insert integers into a statement containing brackets in order to obtain as many different results as they can. This activity can be used to introduce, consolidate or revise simple expanding brackets…
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Intermediate Algebra - Graphing Calc. Manual - 3rd edition
Summary: The Graphing Calculator Manual by Judith A. Penna contains keystroke level instruction for the Texas Instruments TI-83/83+, TI-84, and TI-86. Bundled with every copy of the text, the Graphing Calculator Manual uses actual examples and exercises from Intermediate Algebra: Graphs and Models, Third Edition, to help teach students to use their graphing calculator. The order of topics in the Graphing Calculator Manual mirrors that of the text, providing a just-in-time mode of instruction....show more...8816
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This book is meant to be easily readable to engineers and scientists while still being (almost) interesting enough for mathematics students. Be advised that in-depth proofs of such matters as series convergence, uniqueness, and existence will not be given; this fact will appall some and elate others. This book is meant more toward solving or at the very least extracting information out of problems involving partial differential equations. The first few chapters are built to be especially simple to understand so that, say, the interested engineering undergraduate can benefit; however, later on important and more mathematical topics such as vector spaces will be introduced and used.
What follows is a quick intro for the uninitiated, with analogies to ordinary differential equations.
What is a Partial Differential Equation?
Ordinary differential equations (ODEs) arise naturally whenever a rate of change of some entity is known. This may be the rate of increase of a population, the rate of change of velocity, or maybe even the rate at which soldiers die on a battlefield. ODEs describe such changes of discrete entities. Respectively, this may be the capita of a population, the velocity of a particle, or the size of a military force.
More than one entity may be described with more than one ODE. For example, cloth is very often simulated in computer graphics as a grid of particles interconnected by springs, with Newton's law (an ODE) applied to each "cloth particle". In three dimensions, this would result in 3 second order ODEs written and solved for each particle.
Partial differential equations (PDEs) are analogous to ODEs in that they involve rates of change; however, they differ in that they treat continuous media. For example, the cloth could just as well be considered to be some kind of continuous sheet. This approach would most likely lead to only 3 (maybe 4) partial differential equations, which would represent the entire continuous sheet, instead of a set of ODEs for each particle.
This continuum approach is a very different way of looking at things. It may or may not be favorable: in the case of cloth, the resulting PDE system would be too difficult to solve, and so the computer graphics industry goes with a particle based approach (but a prime counterexample is a fluid, which would be represented by a PDE system most of the time).
While PDEs may not be straightforward to solve on a computer, they have a major advantage over ODEs when applicable: it is nearly impossible to gain any analytical insight from a huge system of particles, while a relatively small PDE system can reveal much insight, even if it won't yield an analytic solution.
But PDEs don't strictly describe continuum mechanics. As with anything mathematical, they are what you make of them.
The Character of Partial Differential Equations
The solution of an ODE can be represented as a function of one variable. For example, the position of the Earth may be represented by coordinates with respect to, say, the sun, and each of these coordinates would be functions of time. Note that the effects of other celestial bodies would certainly affect the solution, but it would still be expressible strictly as a function of time.
The solution of a PDE will, in general, depend on more than one variable. An example is a vibrating string: the deflection of the string will depend both on time and which part of the string you're looking at.
The solution of an ODE is called a trajectory. It may be represented graphically by one or more curves. The solution of a PDE, however, could be a surface, a volume, or something else, depending on how many variables are involved and how they're interpreted.
In general, PDEs are complicated to solve. Concepts such as separation of variables or integral transformations tend to work very differently. One significant difficulty is that the solution of a PDE depends very strongly on the initial/boundary conditions (ICs/BCs). An ODE typically yields a general solution, which involves one or more constants which may be determined from one or more ICs/BCs. PDEs, however, do not easily yield such general solutions. A solution method that works for one initial boundary value problem (IBVP) may be useless for a different IBVP.
PDEs tend to be more difficult to solve numerically as well. Most of the time, an ODE can be expressed in terms of its highest order derivative, and can be solved on a computer very easily with knowledge of the ICs (boundary value problems are a little more complicated), using well established and more or less generally applicable methods, such as Runge Kutta (RK). With this in mind, an ODE may be solved quickly by entering the equation and its ICs/BCs into the right application and pressing the solve button. An IBVP for a PDE, however, will typically require its own specialized solution, and it may take much effort to make the solution more than, say, second order accurate.
An Early Example
Many of the concepts of the previous section may be summarized in this example. We won't deal with the PDE just yet.
Consider heat flow along a laterally insulated rod. In other words, the heat is only flowing along the rod but not into the surrounding air. Let's call the temperature of the rod , and let , where is time and represents the position along the rod. As the temperature depends both on time and position along the rod, this is exactly what says. It is the change of the heat distribution over time. See the graphic below to get an idea.
Let's say that the rod has unitless length , and that its initial temperature (again unitless) is known to be . This states the initial condition, which depends on . The function is a simple hump between 0 and 1. Check for yourself with maxima ( or on android): plot2d(sin(x*%pi),[x,0,1])
Let's also say that the temperature is somehow fixed to at both ends of the rod, i.e. at and at . This would result in , which specifies boundary conditions. The BCs state that for all t, at and .
A PDE can be written to describe the situation. This and the IC/BCs form an initial boundary value problem (IBVP). The solution to this IBVP is (with a physical constant taken to be ):
Note that:
It also satisfies the PDE, but (again) that'll come later.
This solution may be interpreted as a surface, it's shown in the figure below with going from to , and going from to . That is, the distribution of heat is changing over time as the heat flows and dissipates.
from to and to .
Surfaces may or may not be the best way to convey information, and in this case a possibly better way to draw the picture would be to graph as a curve at several different choices of , this is portrayed below.
in the domain of interest for various interesting values of .
PDEs are extremely diverse, and their ICs and BCs can radically affect their solution method. As a result, the best (read: easiest) way to learn is by looking at many different problems and how they're solved.
Introductory Topics and Techniques
Parallel Plate Flow: Easy IC
Formulation
As with ODEs, separation of variables is easy to understand and works well whenever it works. For ODEs, we use the substitution rule to allow antidifferentiation, but for PDEs it's a very different process involving letting dependencies pass through the partial derivatives.
A fluid mechanics example will be used.
Consider two plates parallel to each other of huge extent, separated by a distance of 1. Fluid is smoothly flowing between these two plates in only one direction (call it x). This may be seen in the picture below.
Visualization of the parallel plate flow problem.
After some assumptions, the following PDE may be obtained to describe the fluid flow:
This linear PDE is the result of simplifying the Navier-Stokes equations, a large nonlinear PDE system which describes fluid flow. u is the velocity of the fluid in the x direction, ρ is the density of the fluid, ν is the kinematic viscosity (metaphorically speaking, how hard the molecules of the fluid rub against each other), and Px is the pressure gradient or pressure gradient vector field. Note that u = u(y, t), there is no dependence on x. In other words, the state of the fluid upstream is no different from the state downstream. Notice that we do not consider turbulences and that the state of the flow varies between the upper plate and the lower plate as the speed u is 0 close to the plates.
u(y, t) is a velocity profile. Fluid mechanics typically is concerned with velocity fields, contrary to rigid body mechanics in which the position of an object is what is important. In other words, with a rigid object all 'points' of the object move at the same speed. With a fluid we get a velocity field and each point is moving at its own speed.
The ratio Px/ρ describes the driving force; it's a pressure change (gradient) along the x direction. If Px is negative, then the pressure downstream (positive x) is smaller than the pressure upstream (negative x) and the fluid will flow left to right, i.e., u(y, t) will generally be positive.
Now on to create a specific problem: let's say that a constant negative pressure gradient was applied for a long time, until the velocity profile was steady (steady means "not changing with time"). Then the pressure gradient is suddenly removed, and without this driving force the fluid will slow down and stop. That is the assumption we are going to use for our example calculations. We assume the flow to be steady and then decaying as the pressure (expressed as Px/ρ) is removed. Hence we can remove the term -Px/ρ from our model as well, see (PDE) below.
Initial flow profile.
Let's say that before the pressure was removed, the velocity profile was u(y, t) = sin(π y). With velocity profile we mean, as the velocities are measured on a cross section of the flow, they form a hump. This would make sense: the friction dictates less motion near the plates (see next paragraph), so we could expect a maximum velocity near the centerline (y = 1/2). This assumed profile isn't really correct, but will serve as an example for now. It's graphed at right in the domain of interest.
Before getting into the math, one more thing is needed: boundary conditions. In this case, the BC is called the no slip condition, which states that the velocity of a fluid at a wall (boundary) is equal to the velocity of the wall. If we weren't making this assumption, the fluid would be moving like a rigid object. Since the velocities of the walls (or plates) in this problem are both zero, the velocity of the fluid must be zero at these two boundaries. The BCs are then u(0, t) = 0 (bottom plate) and u(1, t) = 0 (top plate).
The IBVP is:
Notice that the initial condition IC is the same hump from the first section of this book, just turned sideways. This IC implies that the flow is already flowing when we start our calculations. We are kind of calculating the decay of the flow speed.
Separation
Variables are separated the following way: we assume that , where Y and T are (unknown) functions respectively of y and t. This form is substituted into the PDE:
Using yields:
Look carefully at the last equation: the left side of the equation depends strictly on t, and the right side strictly on y, and they are equal. t may be varied independently of y and they'd still be equal, and y may be varied independently of t and they'd still be equal. This can only happen if both sides are constant. This may be shown as follows:
Taking the derivative of both sides, turns the right-hand-side into a constant, 0. The left-hand-side is left as is. Then taking the integrals of both sides yields:
Integration of the ordinary derivative recovers the left side but leaves the right side a constant. It follows by similarity that Y''/Y is a constant as well.
The constant in question is called the separation constant. We could simply give it a letter, such as A, but a good choice of the constant will make work easier later. In this case the best choice is -k2. This will be justified later (but it should be reemphasized that it may be notated any way you want, assuming it can span the domain).
The variables are now separated. The last two equations are two ODEs which may be solved independently (in fact, the Y equation is an eigenvalue problem), though they both contain an unknown constant. Note that ν was kept for the T equation. This choice makes the solution slightly easier, but is again completely arbitrary.
Rearrange and note that the notion buried in the expression below is that a function is equal to its own derivative, which kind of strikes the Euler number bell.
Then solve.
Here is a solution plucked from Ted Woollett's 'Maxima by Example', Chapter 3, '3.2.3 Exact Solution Using desolve'. Maxima's output is not shown but should be easily reconcilable with what has been written.
(%i1) de:(-%k^2*v*T(t))-('diff(T(t),t)) = 0;
(%i2) gsoln:desolve(de,T(t));
The same goes for our right-hand-side.
And solve.
In Maxima the solution goes like this. When Maxima is asking for 'zero' or 'nonzero' then type 'nonzero;':
(%i1)de:(-%k^2*Y(y))-('diff(Y(y),y,2));
(%i2)atvalue('diff(Y(y),y), y=0, Y(0)*%k )$
(%i3)gsoln:desolve(de,Y(y));
The overall solution will be , still with unknown constants, and as of now the product of Y and T. So we are plugging the partial solutions for and back in:
Note that C1 has been multiplied into C2 and C3, reducing the number of arbitrary constants. Because an unknown constant multiplied by an unknown constant yields still an unknown constant.
The IC or BCs should now be applied. If the IC was applied first, coefficients would be equated and all of the constants would be determined. However, the BCs may or may not have been fulfilled (in this case they would, but you're not generally so lucky). So to be safe, the BCs will be applied first:
So A being zero eliminates the term.
If we took B = 0, the solution would have just been u(y, t) = 0 (often called the trivial solution), which would satisfy the BCs and the PDE but couldn't possibly satisfy the IC. So, we take k = nπ instead, where n is any integer. After applying the BCs we have:
Decaying flow.
Then we need to apply the IC to it. Per the IC from above is:
Per the BC is, see above. Setting those two definitions of equal is:
Since t = 0 as per the IC assumption, is becoming . The equality can only hold if B = 1 and n = 1, allowing us to simply remove those two constants from the function in its BC incarnation. That's it! The complete solution is:
It's worth verifying that the IC, BCs, and PDE are all satisfied by this. Also notice that the solution is a product of a function of t and a function of y. The graph at the right is illustrating this. Observe that the profile is plotted for different values of νt, rather than specifying some ν and graphing different values for t. Remember that v is the kinematic viscosity. Hence the decay of flow speed also depends on it. Looking at the solution, t and ν appear only once and they're multiplying, so it's natural to do this. A dimensionless time could have been introduced from the beginning.
So what happens? The fluid starts with its initial profile and slows down exponentially. Note that with x replaced with y and t replaced with νt, this is exactly the same as the result of heat flow in a rod as shown in the introduction. This is not a coincidence: the PDE for the rod describes diffusion of heat, the PDE for the parallel plates describes diffusion of momentum.
Take a second look at the separation constant, -k2. The square is convenient, without it the solution for Y(y) would have involved square roots. Without the negative sign, the solution would have involved exponentials instead of sinusoids, so the constant would have come out imaginary.
The assumption that u(y, t) = Y(y)T(t) is justified by the physics of the problem: it would make sense that the profile would keep its general shape (due to Y(y)), only it'd get flattened over time as the fluid slows down (due to T(t)).
Quickly recap what we did. First we took a model PDE from Navier-Stokes and simplified it by making an assumption about the pressure being removed. Then we sort of first-stage solved it by separating Y and T. Then we applied the BC (boundary conditions) and the IC (initial condition) yielding our final solution.
Parallel Plate Flow: Realistic IC
The Steady State
The initial velocity profile chosen in the last problem agreed with intuition but honestly came out of thin air. A more realistic development follows.
The problem stated that (to come up with an IC) the fluid was under a pressure difference for some time, so that the flow became steady aka flowing steadily. "Steady" is another way of saying "not changing with time", and "not changing with time" is another way of saying that:
Putting this into the PDE from the previous section:
Independent of , the PDE became an ODE with variables separated and thus we can integrate.
The no slip condition results in the following BCs: at and . We can plug the BC values into the integrated ODE and resolve the Cs.
Inserting the Cs and and simplifying yields:
For the sake of example, take (recall that a negative pressure gradient causes left to right flow). Also note that this is a constant gradient or slope. This gives a parabola which starts at , increases to a maximum of at , and returns to at .
This parabola looks pretty much identical to the sinusoid previously used (you must zoom in to see a difference). However, even more so on the narrow domain of interest, the two are very different functions (look at their taylor expansions, for example). Using the parabola instead of the sine function results in a much more involved solution.
So this derives the steady state flow, which we will use as an improved, realistic IC. Recall that the problem is about a fluid that's initially in motion that is coming to a stop due to the absence of a driving force. The IBVP (Initial Boundary Value Problem) is now subtly different:
Separation
Since the only difference from the problem in the last section is the IC, the variables may be separated and the BCs applied with no difference, giving:
But now we're stuck (after applying the BCs)! Applying the IC makes the term go away as t = 0, which is the IC. However, then the IC function can't be made to match:
What went wrong? It was the assumption that . The fact that the IC couldn't be fulfilled means that the assumption was wrong. It should be apparent now why the IC was chosen to be in the previous section.
We can proceed however, thanks to the linearity of the problem. Another detour is necessary, it gets long.
Linearity (the superposition principle specifically) says that if is a solution to the BVP (not the whole IBVP, only the BVP, Boundary Value Problem, the BCs applied) and so is another , then a linear combination, , is also a solution.
Let's take a step back and suppose that the IC was
This is no longer a realistic flow problem but it contains the first two terms of what is called a Fourier sine expansion, see these examples of Fourier sine expansions. We are going to generalize this below. Let's now use this expression and equate it to the half way solution (BCs applied) with being eliminated as t = 0:
And it still can't match. However, observe that the individual terms in the IC can. We simply set the constants to values making both sides match:
Note the subscripts are used to identify each term: they reflect the integer from the separation constant. Solutions may be obtained for each individual term of the IC, identified with :
Linearity states that the sum of these two solutions is also a solution to the BVP (no need for new constants):
So we added the solutions and got a new solution... what is this good for? Try setting :
Each component solution satisfies the BVP, and the sum of these just happened to satisfy our surrogate IC. The IBVP with IC is now solved. It would work the same way for any linear combination of sine functions whose half frequencies are . "Linear combination" means a sum of terms, each multiplied by a constant. The sum is assumed to converge and be term by term differentiable.
Let's do what we just did in a more generalized fashion. First, we make our IC a linear combination of sines (with eliminated as t = 0), in fact, infinitely many of them. But each successive term has to 'converge', it can't stray wildly all over the place.
Second, find the n and B for each term assuming t = 0 (the IC), then plug them back into each term making no assumptions about t, leaving t as is.
Third, sum up all the terms with their individual n and Bs.
Fourth, plug t = 0 into the sum of terms and recover the IC from the first step.
So we went full circle on this example but found the n and Bs because we were able to equate/satisfy each term with the IC. Now we can solve the problem if the IC is a linear combination of sine functions. But the IC for this problem isn't such a sum, it's just a stupid parabola. Or is it?
Series Construction
In the 19th century, a man named Joseph Fourier took a break from helping Napoleon take over the world to ask an important question while studying this same BVP (concerning heat flow): can a function be expressed as a sum of sinusoids, similar to a taylor series? The short answer is yes, if a few reasonable conditions apply as we have already indicated. The long answer follows, and this section is a longer answer.
A function meeting certain criteria may indeed be expanded into a sum of sines, cosines, or both. In our case, all that is needed to accomplish this expansion is to find the coefficients . A little trick involving an integral makes this possible.
The sine function has a very important property called orthogonality. There are many flavors of this, which will be served in the next chapter. Relevant to this problem is the following:
A quick hint may help. Orthogonality literally means two lines at a right angle to each other. These lines could be vectors, each with its own tuple of coordinates. If those two vectors are at a right angle to each other, multiplying and summing their coordinate tuples always yields zero (in Euclidean space). The method of multiplying and summing is also used to determine whether two functions are orthogonal. Using this definition, our multiplied and integrated functions above are orthogonal most of the time, but not always.
Let's call the IC to generalize it. We equate the IC with its expansion, meaning the linear combination of sines, and then apply some craftiness. And remember that our goal is to reproduce a parabolic function from linearly combined sines:
In the last step, all of the terms in the sum became except for the term where , the only case where we get for the otherwise orthogonal sine functions. This isolates and explicitly defines which is the same as as m = n. The expansion for is then:
Or equivalently:
Many important details have been left out for later in a devoted chapter; one noteworthy detail is that this expansion is only approximating the parabola (very superficially) on the interval , not say from to .
This expansion may finally be combined with the sum of sines solution to the BVP developed previously. Note that the last equation looks very similar to . Following from this:
So the expansion will satisfy the IC given as (surprised?). The full solution for the problem with arbitrary IC is then:
In this problem specifically, the IC is , so:
Sines and cosines appear from the integration dependent only on . Since is an integer, these can be made more aesthetic.
Note that for even , . Putting everything together finally completes the solution to the IBVP:
There are many interesting things to observe. To begin with, is not a product of a function of and a function of . Such a solution was assumed in the beginning, proved to be wrong, but eventually happened to yield a solution anyway thanks to linearity and what is called a Fourier sine expansion.
A careful look at the procedure reveals something that may be disturbing: this lengthy solution is strictly valid for the given BCs. Thanks to the definition of , the solution is generic as far as the IC is concerned (the IC doesn't even need to match the BCs), however a slight change in either BC would mandate starting over almost from the beginning.
The parabolic IC, which looks very similar to the sine function used in the previous section, is wholly to blame (or thank once you understand the beauty of a Fourier series!) for the infinite sum. It is interesting to approximate the first several numeric values of the sequence :
Recall that the even terms are all . The first term by far dominates, this makes sense since the first term already looks very, very similar to the parabola. Recall that appears in an exponential, making the higher terms even smaller for time not too close to .
Change of Variables
As with ODEs, a PDE (or more accurately, the IBVP as a whole) may be made more amenable with the help of some kind of modification of variables. So far, we've dealt only with boundary conditions that specify the value of u, which represented fluid velocity, as zero at the boundaries. Though fluid mechanics can get more complicated than that (understatement of the millennium), let's look at heat transfer now for the sake of variety.
As hinted previously, the one dimensional diffusion equation can also describe heat flow in one dimension. Think of how heat could flow in one dimension: one possibility is a rod that's completely laterally insulated, so that the heat will flow only along the rod and not across it (be aware, though, it is possible to consider heat loss/gain along the rod without going two dimensional).
If this rod has finite length, heat could flow in and out of the uninsulated ends. A 1D rod can have at most two ends (it can also have one or zero: the rod could be modeled as "very long"), and the boundary conditions could specify what happens at these ends. For example, the temperature could be specified at a boundary, or maybe the flow of heat, or maybe some combination of the two.
The equation for heat flow is usually given as:
Which is the same as the equation for parallel plate flow, only with ν replaced with α and y replaced with x.
Fixed Temperatures at Boundaries
Let's consider a rod of length 1, with temperatures specified (fixed) at the boundaries. The IBVP is:
φ(x) is the temperature at t = 0. Look at what the BCs say: For all time, the temperature at x = 0 is u0 and at x = 1 is u1. Note that this could be just as well a parallel plate problem: u0 and u1 would represent wall velocities.
The PDE is easily separable, in basically the same way as in previous chapters:
Now, substitute the BCs:
We can't proceed. Among other things, the presence of t in the exponential factor (previously divided out) prevents anything from coming out of this.
This is another example of the fact that the assumption that u(x, t) = X(x)T(t) was wrong. The only thing that prevents us from getting a solution would be the non-zero BCs. This is where changing variables will help: a new variable v(x, t) will be defined in terms of u which will be separable.
Think of how v(x, t) could be defined to make its BCs zero ("homogeneous"). One way would be:
This form is inspired from the appearance of the BCs, and it can be readily seen:
If h(0) = u0 and h(1) = u1, v(x, t) would indeed have zero BCs. Pretty much any choice of h(x) satisfying these conditions would do it, but only one is the best choice. Making the substitution into the PDE:
So now the PDE has been messed up by the new term involving h. This will thwart separation...
...unless that last term happens to be zero. Rather then hoping it's zero, we can demand it (the best choice hinted above), and put the other requirements on h(x) next to that:
Note that the partial derivative became an ordinary derivative since h is a function of x only. The above constitutes a pretty simple boundary value problem, with unique solution:
It's just a straight line. Note that this is what would arise if the steady state (time independent) problem were solved for u(x). In other words, h could've been pulled out of one's ass readily just looking at the physics of the situation.
But anyway. The problem now reduces to finding v(x, t). The IBVP for this would be:
Note that the IC changed under the transformation. The solution to this IBVP was found in a past chapter through separation of variables and superposition to be:
u(x, t) may now be found simply by adding h(x), according to how the variable change was defined:
This solution looks like the sum of a steady state portion (that's h(x)) and a transient portion (that's v(x)):
Visualization of the change of variables.
Time Varying Temperatures at Boundaries
Note that this wouldn't work so nicely with non-constant BCs. For example, if the IBVP were:
Which doesn't really make anything simpler, despite freedom in the choice of IC.
But this isn't completely useless. Note that the PDE for h was chosen to simplify the PDE for v(x, t) (would lead to the terms involving h to cancel out), which may lead to the question: Was this necessary?
The answer is no. If that were the case, the PDE we picked for h would not be satisfied, and that would result in extra terms in the PDE for v(x, t). The no-longer-separable IBVP for v(x, t) could, however, be solved via an eigenfunction expansion, whose full story will be told later. It's worth noting though, that an eigenfunction expansion would require homogenous BCs, so the transformation was necessary.
So this problem has to be put aside without any conclusion for now. I told you that BCs can mess everything up.
Pressure Driven Transient Parallel Plate Flow
Now back to fluid mechanics. Previously, we dealt with flow that was initially moving but slowing down because of resistance and the absence of a driving force. Maybe, it'd be more interesting if we had a fluid initially at rest (ie, zero IC) but set into motion by some constant pressure difference. The IBVP for such a case would be:
This PDE with the pressure term was described previously. That pressure term is what drives the flow; it is assumed constant.
The intent of the change of variables would be to remove the pressure term from the PDE (which prevents separation) while keeping the BCs homogeneous.
One path to take would be to add something to u(x, t), either a function of t or a function of y, so that differentiation would leave behind a constant that could cancel the pressure term out. Adding a function of t would be very unfavorable since it'd result in time dependent BCs, so let's try a function of y:
Substituting this into the PDE:
This procedure will simplify the PDE and preserve the BCs only if the following conditions hold:
The first condition, an ODE, is required to simplify the PDE for v(y, t), it will result in cancellation of the last two terms. The other two conditions are chosen to preserve the homogeneous BCs of the problem (note that if the BCs of u(y, t) weren't homogeneous, the BCs on f(y) would need to be picked to amend that).
The solution to the BVP above is simply:
So f(y) was successfully determined. Note that this function is symmetric about y = 1/2. The IBVP for v(y, t) becomes:
This is the same IBVP we've been beating to death for some time now. The solution for v(y, t) is:
And the solution for u(y, t) follows from how the variable change was defined:
This solution fits what we expect: it starts flat and approaches the parabolic profile quickly. This is the same parabola derived as the steady state flow in the realistic IC chapter; the integral was evaluated for integer n, simplifying it.
A careful look at the solution reveals something interesting: this is just decaying parallel plate flow "in reverse". Instead of the flow starting parabolic and gradually approaching u = 0, it starts with u = 0 and gradually approaches a parabola.
Time Dependent Diffusivity
In this example we'll change time, an independent variable, instead of changing the dependent variable. Consider the following IBVP:
Note that this is a separable; a transformation isn't really necessary, however it'll be easier since we can reuse past solutions if it can be transformed into something familiar.
Let's not get involved with the physics of this and just call it a diffusion problem. It could be diffusion of momentum (as in fluid mechanics), diffusion of heat (heat transfer), diffusion of a chemical (chemistry), or simply a mathematician's toy. In other words, a confession: it was purposely made up to serve as an example.
The (time dependent) factor in front of the second derivative is called the diffusivity. Previously, it was a constant α (called "thermal diffusivity") or constant ν ("kinematic viscosity"). Now, it decays with time.
To simplify the PDE via a transformation, we look for ways in which the factor could cancel out. One way would be to define a new time variable, call it τ and leave it's relation to t arbitrary. The chain rule yields:
Substituting this into the PDE:
Note now that the variable t will completely disappear (divide out in this case) from the equation if:
C is completely arbitrary. However, the best choice of C is the one that makes τ = 0 when t = 0, since this wouldn't change the IC which is defined at t = 0; so, take C = 0. Note that the BCs wouldn't change either way, unless they were time dependent, in which they would change no matter what C is chosen. The IBVP is turned into:
Digging up the solution and restoring the original variable:
Note that, unlike any of the previous examples, the physics of the problem (if there were any) couldn't have helped us. It's also worth mentioning that the solution doesn't limit to u = 0 for long time.
Concluding Remarks
Changing variables works a little differently for PDEs in the sense that you have a lot of freedom thanks to partial differentiation. In this chapter, we picked what seemed to be a good general form for the transformation (inspired by whatever prevented easy solution), wrote down a bunch of requirements, and defined the transformation to uniquely satisfy the requirements. Doing the same for ODEs can often degrade to a monkey with typewriter situation.
Many simple little changes go without saying. For example, we've so far worked with rods of length "1" or plates separated by a distance of "1". What if the rod was 5 m long? Then space would have to be nondimensionalized using the following transformation:
Simple nondimensionalization is, well, simple; however for PDEs with more terms it can lead to scale analysis which can lead to perturbation theory which will all have to be explained in a later chapter.
It's worth noting that the physics of the IBVP very often suggest what kind of transformation needs to be done. Even some nonlinear problems can be solved this way.
This topic isn't nearly over, changes of variables will be dealt with again in future chapters.
The Laplacian and Laplace's Equation
By now, you've most likely grown sick of the one dimensional transient diffusion PDE we've been playing with:
Make no mistake: we're not nearly done with this stupid thing; but for the sake of variety let's introduce a fresh new equation and, even though it's not strictly a separation of variables concept, a really cool quantity called the Laplacian. You'll like this chapter; it has many pretty pictures in it.
Graph of .
The Laplacian
The Laplacian is a linear operator in Euclidean n-space. There are other spaces with properties different from Euclidean space. Note also that operator here has a very specific meaning. As a function is sort of an operator on real numbers, our operator is an operator on functions, not on the real numbers. See here for a longer explanation.
We'll start with the 3D Cartesian "version". Let . The Laplacian of the function is defined and notated as:
So the operator is taking the sum of the nonmixed second derivatives of with respect to the Cartesian space variables . The "del squared" notation is preferred since the capital delta can be confused with increments and differences, and is too long and doesn't involve pretty math symbols. The Laplacian is also known as the Laplace operator or Laplace's operator, not to be confused with the Laplace transform. Also, note that if we had only taken the first partial derivatives of the function , and put them into a vector, that would have been the gradient of the function . The Laplacian takes the second unmixed derivatives and adds them up.
In one dimension, recall that the second derivative measures concavity. Suppose ; if is positive, is concave up, and if is negative, is concave down, see the graph below with the straight up or down arrows at various points of the curve. The Laplacian may be thought of as a generalization of the concavity concept to multivariate functions.
This idea is demonstrated at the right, in one dimension: . To the left of , the Laplacian (simply the second derivative here) is negative, and the graph is concave down. At , the curve inflects and the Laplacian is . To the right of , the Laplacian is positive and the graph is concave up.
Concavity may or may not do it for you. Thankfully, there's another very important view of the Laplacian, with deep implications for any equation it shows itself in: the Laplacian compares the value of at some point in space to the average of the values of in the neighborhood of the same point. The three cases are:
If is greater at some point than the average of its neighbors, .
If is at some point equal to the average of its neighbors, .
If is smaller at some point than the average of its neighbors, .
So the laplacian may be thought of as, at some point :
The neighborhood of .
The neighborhood of some point is defined as the open set that lies within some Euclidean distance δ (delta) from the point. Referring to the picture at right (a 3D example), the neighborhood of the point is the shaded region which satisfies:
With this mentality, let's examine the behavior of this very important PDE. On the left is the time derivative and on the right is the Laplacian. This equation is saying that:
The rate of change of at some point is proportional to the difference between the average value of around that point and the value of at that point.
For example, if there's at some position a "hot spot" where is on average greater then its neighbors, the Laplacian will be negative and thus the time derivative will be negative, this will cause to decrease at that position, "cooling" it down. This is illustrated below. The arrows reflect upon the magnitude of the Laplacian and, by grace of the time derivative, the direction the curve will move.
Visualization of transient diffusion.
It's worth noting that in 3D, this equation fully describes the flow of heat in a homogeneous solid that's not generating it's own heat (like too much electricity through a narrow wire would).
Laplace's Equation
Laplace's equation describes a steady state condition, and this is what it looks like:
Solutions of this equation are called harmonic functions. Some things to note:
Time is absent. This equation describes a steady state condition.
The absence of time implies the absence of an IC, so we'll be dealing with BVPs rather then IBVPs.
In one dimension, this is the ODE of a straight line passing through the boundaries at their specified values.
All functions that satisfy this equation in some domain are analytic (informally, an analytic function is equal to its Taylor expansion) in that domain.
Laplace's equation is separable in the Cartesian (and almost any other) coordinate system. So, we shouldn't have too much problem solving it if the BCs involved aren't too convoluted.
Laplace's Equation on a Square: Cartesian Coordinates
Steady state conditions on a square.
Imagine a 1 x 1 square plate that's insulated top and bottom and has constant temperatures applied at its uninsulated edges, visualized to the right. Heat is flowing in and out of this thing steadily through the edges only, and since it's "thin" and "insulated", the temperature may be given as . This is the first time we venture into two spatial coordinates, note the absence of time.
Let's make up a BVP, referring to the picture:
So we have one nonhomogeneous BC. Assume that :
As with before, calling the separation constant in favor of just (or something) happens to make the problem easier to solve. Note that the negative sign was kept for the equation: again, these choices happen to make things simpler. Solving each equation and combining them back into :
At edge D:
Note that the constants can be merged, but we won't do it so that a point can be made in a moment. At edge A:
Taking as would satisfy this particular BC, however this would yield a plane solution of , which can't satisfy the temperature at edge C. This is why the constants weren't merged a few steps ago, to make it obvious that may not be . So, we instead take to satisfy the above, and then combine the three constants into one, call it :
Now look at edge B:
It should go without saying by now that can't be zero, since this would yield which couldn't satisfy the nonzero BC. Instead, we can take :
As of now, this solution will satisfy 3 of the 4 BCs. All that is left is edge C, the nonhomogeneous BC.
Neither nor can be contorted to fit this BC.
Since Laplace's equation is linear, a linear combination of solutions to the PDE is also a solution to the PDE. Another thing to note: since the BCs (so far) are homogeneous, we can add the solutions without worrying about nonzero boundaries adding up.
Though as shown above will not solve this problem, we can try summing (based on ) solutions to form a linear combination which might solve the BVP as a whole:
It looks like it needs Fourier series methodology. Finding via orthogonality should solve this problem:
25 term partial sum of the series solution.
was changed to in the last step. Also, for integer , . Note that a Fourier sine expansion has been done. The solution to the BVP can finally be assembled:
That solves it!
It's finally time to mention that the BCs are discontinuous at the points and . As a result, the series should converge slowly at those points. This is clear from the plot at right: it's a 25 term partial sum (note that half of the terms are ), and it looks perfect except at , especially near the discontinuities at and .
Laplace's Equation on a Circle: Polar Coordinates
Now, we'll specify the value of on a circular boundary. A circle can be represented in Cartesian coordinates without too much trouble; however, it would result in nonlinear BCs which would render the approach useless. Instead, polar coordinates should be used, since in such a system the equation of a circle is very simple. In order for this to be realized, a polar representation of the Laplacian is necessary. Without going in to the details just yet, the Laplacian is given in (2D) polar coordinates:
This result may be derived using differentials and the chain rule; it's not difficult but it's a little long. In these coordinates Laplace's equation reads:
Note that in going from Cartesian to polar coordinates, a price was paid: though still linear, Laplace's equation now has variable coefficients. This implies that after separation at least one of the ODEs will have variable coefficients as well.
Let's make up the following BVP, letting :
This could represent a physical problem analogous to the previous one: replace the square plate with a disc. Note the apparent absence of sufficient BC to obtain a unique solution. The funny looking statement that u is bounded inside the domain of interest turns out to be the key to getting a unique solution, and it often shows itself in polar coordinates. It "makes up" for the "lack" of BCs. To separate, we as usual incorrectly assume that :
Once again, the way the negative sign and the separation constant are arranged makes the solution easier later on. These decisions are made mostly by trial and error.
The equation is probably one you've never seen before, it's a special case of the Euler differential equation (not to be confused with the Euler-Lagrange differential equation). There are a couple of ways to solve it, the most general method would be to change the variables so that an equation with constant coefficients is obtained. An easier way would be to note the pattern in the order of the coefficients and the order of the derivatives, and from there guess a power solution. Either way, the general solution to this simple case of Euler's ODE is given as:
This is a very good example problem since it goes to show that PDE problems very often turn into obscure ODE problems; we got lucky this time since the solution for was rather simple though its ODE looked pretty bad at first sight. The solution to the equation is:
Combining:
Now, this is where the English sentence condition stating that u must be bounded in the domain of interest may be invoked. As , the term involving is unbounded. The only way to fix this is to take . Note that if this problem were solved between two concentric circles, this term would be nonzero and very important. With that term gone, constants can be merged:
Only one condition remains: on , yet there are 3 constants. Let's say for now that:
Then, it's a simple matter of equating coefficients to obtain:
Now, let's make the frequencies differ:
Equating coefficients won't work. However, if the IC were broken up into individual terms, the sum of the solution to the terms just happens to solve the BVP as a whole:
Verify that the solution above is really equal to the BC at :
And, since Laplace's equation is linear, this must solve the PDE as well. What all of this implies is that, if some generic function may be expressed as a sum of sinusoids with angular frequencies given by , all that is needed is a linear combination of the appropriate sum. Notated:
To identify the coefficients, substitute the BC:
The coefficients and may be determined by a (full) Fourier expansion on . Note that it's implied that must have period since we are solving this in a domain (a circle specifically) where .
You probably don't like infinite series solutions. Well, it happens that through a variety of manipulations it's possible to express the full solution of this particular problem as:
This is called Poisson's integral formula.
Derivation of the Laplacian in Polar Coordinates
Though not necessarily a PDEs concept, it is very important for anyone studying this kind of math to be comfortable with going from one coordinate system to the next. What follows is a long derivation of the Laplacian in 2D polar coordinates using the multivariable chain rule and the concept of differentials. Know, however, that there are really many ways to do this.
Three definitions are all we need to begin:
If it's known that , then the chain rule may be used to express derivatives in terms of and alone. Two applications will be necessary to obtain the second derivatives. Manipulating operators as if they meant something on their own:
Applying this to itself, treating the underlined bit as a unit dependent on and :
The above mess may be quickly simplified a little by manipulating the funny looking derivatives:
This may be made slightly easier to work with if a few changes are made to the way some of the derivatives are written. Also, the variable follows analogously:
Now we need to obtain expressions for some of the derivatives appearing above. The most direct path would use the concept of differentials. If:
Then:
Solving by substitution for and gives:
If , then the total differential is given as:
Note that the two previous equations are of this form (recall that and , just like above), which means that:
Equating coefficients quickly yields a bunch of derivatives:
There's an easier but more abstract way to obtain the derivatives above that may be overkill but is worth mentioning anyway. The Jacobian of the functions and is:
Note that the Jacobian is a compact representation of the coefficients of the total derivative; using as an example (bold indicating vectors):
So, it follows then that the derivatives that we're interested in may be obtained by inverting the Jacobian matrix:
Though somewhat obscure, this is very convenient and it's just one of the many utilities of the Jacobian matrix. An interesting bit of insight is gained: coordinate changes are senseless unless the Jacobian is invertible everywhere except at isolated points, stated another way the determinant of the Jacobian matrix must be nonzero, otherwise the coordinate change is not one-to-one (note that the determinant will be zero at in this example. An isolated point such as this is not problematic.).
Either path you take, there should now be enough information to evaluate the Cartesian second derivatives. Working on :
Proceeding similarly for :
Now, add these tirelessly hand crafted differential operators and watch the result collapse into just 3 nontrigonometric terms:
That was a lot of work. To save trouble, here is the Laplacian in other two other popular coordinate systems:
Fundamentals
Introduction and Classifications
The intent of the prior chapters was to provide a shallow introduction to PDEs and their solution without scaring anyone away. A lot of fundamentals and very important details were left out. After this point, we are going to proceed with a little more rigor; however, knowledge past one undergraduate ODE class alongside some set theory and countless hours on Wikipedia should be enough.
Some Definitions and Results
An equation of the form
is called a partial differential equation if is unknown and the function involves partial differentiation. More concisely, is an operator or a map which results in (among other things) the partial differentiation of . is called the dependent variable, the choice of this letter is common in this context. Examples of partial differential equations (referring to the definition above):
Note that what exactly is made of is unspecified, it could be a function, several functions bundled into a vector, or something else; but if satisfies the partial differential equation, it is called a solution. If it doesn't, everyone will laugh at you.
Another thing to observe is seeming redundancy of , its utility draws from the study of linear equations. If , the equation is called homogeneous, otherwise it's nonhomogeneous or inhomogeneous.
It's worth mentioning now that the terms "function", "operator", and "map" are loosely interchangeable, and that functions can involve differentiation, or any operation. This text will favor, not exclusively, the term function.
The order of a PDE is the order of the highest derivative appearing, but often distinction is made between variables. For example the equation
is second order in and fourth order in (fourth derivatives will result regardless of the form of ).
Linear Partial Differential Equations
Suppose that , and that satisfies the following properties:
for any scalar . The first property is called additivity, and the second one is called homogeneity. If is additive and homogeneous, it is called a linear function, additionally if it involves partial differentiation and
then the equation above is a linear partial differential equation. This is where the importance of shows up. Consider the equation
where is not a function of . Now, if we represent the equation through
then fails both additivity and homogeneity and so is nonlinear (Note: the equation defining the condition is 'homogeneous', but in a distinct usage of the term). If instead
then is now linear. Note then that the choice of and is generally not unique, but if an equation could be written in a linear form it is called a linear equation.
Linear equations are very popular. One of the reasons for this popularity is a little piece of magic called the superposition principle. Suppose that both and are solutions of a linear, homogeneous equation (here onwards, will denote a linear function), ie
for the same . We can feed a combination of and into the PDE and, recalling the definition of a linear function, see that
for some constants and . As stated previously, both and are solutions, which means that
What all this means is that if both and solve the linear and homogeneous equation , then the quantity is also a solution of the partial differential equation. The quantity is called a linear combination of and . The result would hold for more combinations, and generally,
The Superposition Principle
Suppose that in the equation
the function is linear. If some sequence satisfies the equation, that is if
then any linear combination of the sequence also satisfies the equation:
where is a sequence of constants and the sum is arbitrary.
Note that there is no mention of partial differentiation. Indeed, it's true for any linear equation, algebraic or integro-partial differential-whatever. Concerning nonhomogeneous equations, the rule can be extended easily. Consider the nonhomogeneous equation
Let's say that this equation is solved by and that a sequence solves the "associated homogeneous problem",
where is the same between the two. An extension of superposition is observed by, say, the specific combination :
More generally,
The Extended Superposition Principle
Suppose that in the nonhomogeneous equation
the function is linear. Suppose that this equation is solved by some , and that the associated homogeneous problem
is solved by a sequence . That is,
Then plus any linear combination of the sequence satisfies the original (nonhomogeneous) equation:
where is a sequence of constants and the sum is arbitrary.
The possibility of combining solutions in an arbitrary linear combination is precious, as it allows the solutions of complicated problems be expressed in terms of solutions of much simpler problems.
This part of is why even modestly nonlinear equations pose such difficulties: in almost no case is there anything like a superposition principle.
Classification of Linear Equations
A linear second order PDE in two variables has the general form
If the capital letter coefficients are constants, the equation is called linear with constant coefficients, otherwise linear with variable coefficients, and again, if = 0 the equation is homogeneous. The letters and are used as generic independent variables, they need not represent space. Equations are further classified by their coefficients; the quantity
is called the discriminant. Equations are classified as follows:
Note that if coefficients vary, an equation can belong to one classification in one domain and another classification in another domain. Note also that all first order equations are parabolic.
Smoothness of solutions is interestingly affected by equation type: elliptic equations produce solutions that are smooth (up to the smoothness of coefficients) even if boundary values aren't, parabolic equations will cause the smoothness of solutions to increase along the low order variable, and hyperbolic equations preserve lack of smoothness.
Generalizing classifications to more variables, especially when one is always treated temporally (ie associated with ICs, but we haven't discussed such conditions yet), is not too obvious and the definitions can vary from context to context and source to source. A common way to classify is with what's called an elliptic operator.
Definition: Elliptic Operator
A second order operator of the form
is called elliptic if , an array of coefficients for the highest order derivatives, is a positive definite symmetric matrix. is the imaginary unit. More generally, an order elliptic operator is
if the dimensional array of coefficients of the highest () derivatives is analogous to a positive definite symmetric matrix.
Not commonly, the definition is extended to include negative definite matrices.
The negative of the Laplacian, , is elliptic with . The definition for the second order case is separately provided because second order operators are by a large margin the most common.
Classifications for the equations are then given as
for some constant k. The most classic examples of these equations are obtained when the elliptic operator is the Laplacian: Laplace's equation, linear diffusion, and the wave equation are respectively elliptic, parabolic, and hyperbolic and are all defined in an arbitrary number of spatial dimensions.
Other classifications
Quasilinear
The linear form
was considered previously with the possibility of the capital letter coefficients being functions of the independent variables. If these coefficients are additionally functions of which do not produce or otherwise involve derivatives, the equation is called quasilinear. It must be emphasized that quasilinear equations are not linear, no superposition or other such blessing; however these equations receive special attention. They are better understood and are easier to examine analytically, qualitatively, and numerically than general nonlinear equations.
A common quasilinear equation that'll probably be studied for eternity is the advection equation
which describes the conservative transport (advection) of the quantity in a velocity field . The equation is quasilinear when the velocity field depends on , as it usually does. A specific example would be a traffic flow formulation which would result in
Despite resemblance, this equation is not parabolic since it is not linear. Unlike its parabolic counterparts, this equation can produce discontinuities even with continuous initial conditions.
General Nonlinear
Some equations defy classification because they're too abnormal. A good example of an equation is the one that defines a minimal surface expressible as :
Vector Spaces: Mathematic Playgrounds
The study of partial differential equations requires a clear definition of what kind of numbers are being dealt with and in what way. PDEs are normally studied in certain kinds of vector spaces, which have a number of properties and rules associated with them which make possible the analysis and unifies many notions.
The Real Field
A field is a set that is bundled with two operations on the set called addition and multiplication which obey certain rules, called axioms. The letter will be used to represent the field, and from definition a field requires the following ( and are in ):
Closure under addition and multiplication: the addition and multiplication of field members produces members of the same field.
Addition and multiplication are associative: and .
Addition and multiplication are commutative: and .
Addition and multiplication are distributive: and .
Existence of additive identity: there is an element in notated 0, sometimes called the sum of no numbers, such that .
Existence of multiplicative identity: there is an element in notated 1 different from 0, sometimes called the product of no numbers, such that .
Existence of additive inverse: there is an element in associated with notated such that .
Existence of multiplicative inverse: there is an element in associated with (if is nonzero), notated such that .
These are called the field axioms. The field that we deal with, by far the most common one, is the real field. The set associated with the real field is the set of real numbers, and addition and multiplication are the familiar operations that everyone knows about.
Another example of a set that can form a field is the set of rational numbers, numbers which are expressible as the ratio of two integers. An example of a common set that doesn't form a field is the set of integers: there generally is no multiplicative inverse since the reciprocal of an integer generally is not an integer.
Note that when we say that an object is in, what is meant is that the object is a member of the set associated in the field and that it complies with the field axioms.
The Vector
Most non-mathematics students are taught that vectors are ordered groups ("tuples") of quantities. This is not complete, vectors are a lot more general than that. Informally, a vector is defined as an object that can be scaled and added with other vectors. This will be made more specific soon.
The integers, at least when scaled by real numbers (since the result will not necessarily be an integer).
An interesting (read: confusing) fact to note is that, by the definition above, matrices and even tensors qualify as vectors since they can be scaled or added, even though these objects are considered generalizations of more "conventional" vectors, and calling a tensor a vector will lead to confusion.
The Vector Space
A vector space can be thought of as a generalization of a field.
Letting represent some field, a vector space over is a set of vectors bundled with two operations called vector addition and scalar multiplication, notated:
Vector addition: , where .
Scalar multiplication: , where and .
The members of are called vectors, and the members of the field associated with are called scalars. Note that these operations imply closure (see the first field axiom), so that it does not have to be explicitly stated. Note also that this is essentially where a vector is defined: objects that can be added and scaled. The vector space must comply with the following axioms ( and are in ; and are in ):
Addition is associative: .
Addition is commutative: .
Scalar multiplication is distributive over vector addition: .
Scalar multiplication is distributive over field addition: .
Scalar and field multiplication are compatible: .
Existence of additive identity: there is an element in notated 0 such that .
Existence of additive inverse: there is an element in associated with notated such that .
Existence of multiplicative identity: there is an element in notated 1 different from 0 such that .
An example of a vector space is one where polynomials are vectors over the real field. An example of a space that is not a vector space is one where vectors are rational numbers over the real field, since scalar multiplication can lead to vectors that are not rational (implied closure under scalar multiplication is violated).
By analogy with linear functions, vectors are linear by nature, hence a vector space is also called a linear space. The name "linear vector space" is also used, but this is somewhat redundant since there is no such thing as a nonlinear vector space. It's now worth mentioning an important quantity called a linear combination (not part of the definition of a vector space, but important):
where is a sequence of field members and is a sequence of vectors. The fact that a vector can be formed by a linear combination of other vectors is much of the essence of the vector field.
Note that a field over itself qualifies as a vector space. Fields of real numbers and other familiar objects are sometimes called spaces, since distance and other useful concepts apply.
The definition of a vector space is quite general. Note that, for example, there is no mention of any kind of product between vectors, nor is there a notion of the "length" of a vector. The vector space as defined above is very primitive, but it's a starting point: through various extensions, specific vector spaces can have a lot of nice properties and other features that make our playgrounds fun and comfortable. We'll discuss bases (plural of basis) and then take on some specific vector spaces.
The Basis
A nonempty subset of is called a linear subspace of if is itself a vector space. The requirement that be a vector space can be safely made specific by saying that is closed under vector addition and scalar multiplication, since the rest of the vector space properties are inherited.
The linear span of a set of vectors in may then be defined as:
Where . The span is the intersection over all choices of . This concept may be extended so that is not necessarily finite. The span of is the intersection of all of the linear subspaces of .
Now, think of what happens if a vector is removed from the set . Does the span change? Not necessarily, it may be possible that the remaining vectors in the span are sufficient to "fill in" for the missing vector through linear combination of the remaining vectors.
Let be a subset of . If the span of is the same as the span of , and if removing a vector from necessarily changes its span, then the set of vectors is called a basis of , and the vectors of are called linearly independent. It can be proven that a basis can be constructed for every vector space.
Note that the basis is not unique. This obscure definition of a basis is convenient because it is very broad, it is worth understanding fully. An important property of a vector space is that it necessarily has a basis, and that any vector in the space may be written in terms of a linear combination of the members of the basis.
A more understandable (though less elemental) explanation is provided: for a vector space over the field , the vectors form a basis of (where are in and the following are in ), satisfying the following properties:
The basis vectors are linearly independent: if
then
without exception.
The basis vectors span : for some given in , it is possible to choose so that
The basis vectors of a vector space are usually notated with as .
Euclidean n-Space
As most students are familiar with Euclidean n-space, this section serves more of an example than anything else.
Let be the field of real numbers, then the vector space over is defined to be the space of n-tuples of members of . In other words, more clearly:
If , then , where are the vectors of the vector space .
These vectors are called n-dimensional coordinates, and the vector space is called the real coordinate space; note that coordinates (unlike more general vectors) are often notated in boldface, or else with an arrow over the letter. They are also called spatial vectors, geometric vectors, just "vectors" if the context allows, and sometimes "points" as well, though some authors refuse to consider points as vectors, attributing a "fixed" sense to points so that points can't be added, scaled, or otherwise messed with. Part of the reason for this is that it allows one to say that some vector space is bound to a point, the point being called the origin.
The Euclidean n-space is the special real coordinate n-space with some additional structure defined which (finally) gives rise to the geometric notion of (specifically these) vectors.
To begin with, an inner product is first defined, notated with either angle braces or a dot:
This quantity, which turns two vectors into a scalar (a member of ) doesn't have a great deal of geometric meaning until some more structure is defined. In a coordinate space, the dot notation is favored, and this product is often called the "dot product", especially when or . The definition of this inner product qualifies as an inner product space.
Next comes the norm, in terms of the inner product:
The notation involving single pipes around the letter x is common, again, when or , due to analogy with absolute value, for real and especially complex numbers. For a coordinate space, the norm is often called the length of . This quickly leads to the notion of the distance between two vectors:
Which is simply the length of the vector "from" to .
Finally, the angle between and is defined through, for ,
The motivation for this definition of angle, valid for any , comes from the fact that one can prove that the literal measurable angle between two vectors in satisfies the above (the norm is motivated similarly). Discussing these 2D angles and distances of course mandates making precise the notion of a vector as an "arrow" (ie, correlating vectors to things you can draw on a sheet of paper), but that would get involved and most are already subconsciously familiar with this and it's not the point of this introduction.
This completes the definition of . A thorough introduction to Euclidean space isn't very fitting in a text on Partial Differential equations, it is included so that one can see how a familiar vector space can be constructed ground-up through extensions called "structure".
Banach Spaces
Banach spaces are more general than Euclidean space, and they begin our departure from vectors as geometric objects into vectors as toys in the crazy world of functional analysis.
To be terse, a Banach space is defined as any complete normed vector space. The details follow.
The Inner Product
The inner product is a vector operation which results in a scalar. The vectors are members of a vector space , and the scalar is a member of the field associated with . A vector space on which an inner product is defined is said to be "equipped" with an inner product, and the space is an inner product space. The inner product of and is usually notated .
A truly general definition of the inner product would be long. Normally, if the vectors are real or complex in nature (eg, complex coordinates or real valued functions), the inner product must satisfy the following axioms:
Distributive in the first variable: .
Associative in the first variable: .
Nondegeneracy and nonnegativity: , equality will hold only when .
Conjugate symmetry: .
Note that if the space is real, the last requirement (the overbar indicates complex conjugation) simplifies to , and then the first two axioms extend to the second variable.
A desirable property of an inner product is some kind of orthogonality. Two nonzero vectors are said to be orthogonal only if their inner product is zero. Remember that we're talking about vectors in general, not specifically Euclidean.
Inner products are by no means unique, good definitions are what add quality to specific spaces. The Euclidean inner product, for example, defines the Euclidean distance and angle, quantities which form the foundation of Euclidean geometry.
The Norm
The norm is usually, though not universally, defined in terms of the inner product, which is why the inner product was discussed first (to be technically correct, a Banach space doesn't necessarily need to have an inner product). The norm is an operation (notated with double pipes) which takes one vector and generates one scalar, necessarily satisfying the following axioms:
Scalability: .
The triangle inequality: .
Nonnegativity: , equality only when .
The fact that can be proven from the first two statements above.
Definition requires that only when (compare this to the inner product, which can be zero even if fed nonzero vectors); if this condition is relaxed so that is possible for nonzero vectors, the resulting operation is called a seminorm.
The distance between two vectors and is a useful quantity which is defined in terms of the norm:
The distance is often called the metric, and a vector space equipped with a distance is called a metric space.
Completeness
A Cauchy sequence shown in blue.
A sequence that is not Cauchy. The elements of the sequence fail to get close to each other as the sequence progresses.
As stated before, a Banach space is defined as a complete normed vector space. The norm was described above, so that all that is left to establish the definition of a Banach space is completeness.
Consider a sequence of vectors in a vector space . This sequence of vectors is called a Cauchy sequence if these vectors "tend" toward some "destination" vector, as shown in the pictures at right. Stated precisely, a sequence is a Cauchy sequence if it is always possible to make the distance arbitrarily small by picking larger values of and .
The limit of a Cauchy sequence is:
A vector space is called complete if every Cauchy sequence has a limit that is also in . A Banach space is, finally, a vector space equipped with a norm that is complete. Note that completeness implies existence of distance, which means that every Banach space is a metric space.
An example of a vector space that is complete is Euclidean n-space. An example of a vector space that isn't complete is the space of rational numbers over rational numbers: it is possible to form a sequence of rational numbers which limit to an irrational number.
Hilbert spaces
Note that the inner product was defined above but not subsequently used in the definition of a Banach space. Indeed, a Banach space must have a norm but doesn't necessarily need to have an inner product. However, if the norm in a Banach space is defined through the inner product by
then the resulting special Banach space is called a Hilbert space. Hilbert spaces are important in the study of partial differential equations (some relevance finally!) because many theorems and important results are valid only in Hilbert spaces.
Nondimensionalization
Introduction
You may have noticed something possibly peculiar about all of the problems so far dealt with: "simple" numbers like or keep appearing in BCs and elsewhere. For example, we've so far dealt with BCs such as:
Is this meant to simplify this book because the author is lazy? No. Well, actually, the author is lazy, but it's really the result of what's known as nondimensionalization.
On a basic level, nondimensionalization does two things:
Gets all units out of the problem.
Makes relevant variables range from to or so.
The second point has very serious implications which will have to wait for later. We'll talk about getting units out of the problem for now: important because most natural functions don't have any meaning when fed a unit. For example, is a goofy expression which doesn't mean anything at all (consider its Taylor expansion if you're not convinced).
Do not misunderstand: you can solve any problem you like keeping units in place. That's why angular velocity has units of Hz (), so then has meaning if is in seconds (or can be made to be).
A motivation for nondimensionalization can be seen by noting that ratios of variables to dimensions ("dimension" includes both a size and a unit) have a tendency to show up again and again. Examine what happens if steady state parallel plate flow (an ODE) with the walls separated by , not , is solved:
Let's keep the dimensions of and unspecified for now. Solving this BVP:
Note that we have showing up. Not a coincidence; this implies that the dimensionless problem (or at least halfway dimensionless. We haven't discussed the dimensions of ) could be setup by altering the variable:
is the normalized version of , it ranges from to where varies from to . It is said that is scaled by .
This new variable may be substituted into the problem:
Since the new variable contains no unit, it should be obvious that the rational coefficient must have units of velocity if is to have units of velocity. With this in mind, the coefficient may be divided:
We may define another new variable, a nondimensional velocity:
Substituting this into the equation:
It's finally time to ask an important question: Why?
There are many benefits. The original problem involved 4 parameters: viscosity, density, pressure gradient, and wall separation distance. In this fully nondimensionalized solution, there happens to be none such parameters. The shortened equation above completely describes the behavior of the solution, it contains all of the relevant information.
The solution of one nondimensional problem is far more useful then the solution of a specific dimensional problem. This is especially true if the problem only yields a numeric solution: solving the nondimensional problem greatly reduces the number of charts and graphs that need to be made since you've reduced the number of parameters that could affect the solution.
This leads to another important question, and the culmination of this chapter: here, we first solved a generic dimensional problem and then nondimensionalized it. This luxury wouldn't be available with a more complicated problem. Could it have been nondimensionalized beforehand? Yep. Recalling the BVP:
Note that varies from to in the domain we're interested in. For this reason, it is natural to scale with :
Note that we could have just as well scaled with numbers like or e10.0687 D and still ended up with nondimensionalized and everything mathematically sound. However, alone was the best choice since the resulting variable would vary from to . With this choice of scale, the variable is called normalized in addition to being nondimensional; being normalized is a desirable attribute for mathematic simplicity, accurate numeric evaluation, sense of scale, and other reasons.
What about ? The character of was known, the same can't be said for (why are we solving problems in the first place?). Let's come up with a name for the unknown scale of , say , and normalize using this unknown constant:
Using the chain rule, the new variables may be put into the ODE:
So we have our derivative now. It may be substituted into the ODE:
Remember that was some constant pulled out of thin air. Hence, it can be anything we want it to be. To nondimensionalize the equation and simplify it as much as possible, we may pick:
So that the ODE will become:
The BCs are homogeneous, so they simplify easily. Noting that when :
This may now be quickly solved:
So this isn't quite the same as the nondimensional solution developed from the dimensional solution: there's a factor of on the right side. Consequently, is missing the . It's not a problem, both developments solve the problem and nondimensionalize it. Note that in doing this, we got the following result before even solving the BVP:
This tells much about the size of the velocity.
Before closing this chapter, it's worth mentioning that, generally, if and , where , , , and are all constants,
The pieces that make up f(x), f'(x), and f''(x) are continuous over closed subintervals.
The first requirement is most significant; the last two requirements can, to an extent, be partly eased off in most cases without any trouble. An interesting thing happens at discontinuities. Suppose that f(x) is discontinuous at x = a; the expansion will converge to the following value:
So the expansion converges to the average of the values to the left and the right of the discontinuity. This, and the fact that it converges in the first place, is very convenient. The Fourier series looks unfriendly but it's honestly working for you.
The information needed to express f(x) as a Fourier series are the sequences An and Bn. This is done using orthogonality, which for the sinusoids may be derived easily using a few identities. The following are some useful orthogonality relations, with m and n restricted to integers:
δm,n is called the Kronecker delta, defined by:
The Kronecker delta may be thought of as a discrete version of the Dirac delta "function". Relevant to this topic is its sifting property:
Derivation of the Fourier Series
We're now ready to find An and Bn.
This is supposed to hold for an arbitrary integer m. If m = 0, note that the sum doesn't allow n = 0 and so the sum would be zero since in no case does m = n. This leads to:
This secures A0. Now suppose that m > 0. Since m and n are now in the same domain, the Kronecker delta will do its sifting:
In the second to the last step, sin(mπ) = 0 for integer m. In the last step, m was replaced with n. This defines An for n > 0. For the case n = 0,
Which happens to match the previous development (now you know why it's A0/2 and not just A0). So the sequence An is now completely defined for any value of n of interest:
To get Bn, nearly the same routine is used.
The Fourier series expansion of f(x) is now complete. To have it all in one place:
f(x): a square wave.
It's finally time for an example. Let's derive the Fourier series representation of a square wave, pictured at the right:
In the last bit we used the fact that all of the even terms happened to be absent, and the odd numbers are given by 2n - 1 for integer n. The sum will indeed converge to the square wave, except at the discontinuities where it'll converge to zero (the average of 1 and -1).
Graphs of partial sums are shown at right. Note that this particular expansion doesn't converge too quickly, and that as an approximation of the square wave it's poorest near the discontinuities.
There's another interesting thing to note: all of the cosine terms are absent. It's no coincidence, and this may be a good time to introduce the Fourier sine and cosine expansions for, respectively, odd and even functions.
Periodic Extension and Expansions for Even and Odd functions
Two important expansions may be derived from the Fourier expansion: the Fourier sine series and the Fourier cosine series, the first one was used in the previous section. Before diving in, we must talk about even and odd functions.
Suppose that feven(x) is an even function and fodd(x) is an odd function. That is:
Some interesting identities hold for such functions. Relevant ones include:
This is all very relevant to Fourier series. Suppose that an even function is expanded. Recall that sine is odd and cosine is even. Then:
(whole integrand is even)
(whole integrand is odd)
So the Fourier cosine series (note that all sine terms disappear) is just the Fourier series for an even function, given as:
A Fourier expansion may be similarly built for an odd function:
(whole integrand is odd)
(whole integrand is even)
And the Fourier sine series is:
At this point, the periodic extension may be considered. In the previous chapter, the problem mandated a sine expansion of a parabola. A parabola is by no means a periodic function, and yet a Fourier sine expansion was done on it. What actually happened was that the function was expanded as expected within its domain of interest: the interval 0 ≤ x ≤ 1. Inside this interval, the expansion truly is a parabola. Outside this interval, the expansion is periodic, and as a whole is odd (just like the sine functions it's built on).
The parabola could've been expanded just as well using cosines (resulting in an even expansion) or a full Fourier expansion on, say, -1 ≤ x ≤ 1.
Note that we weren't able to pick which expansion to use, however. While the parabola could be expanded any way we want on any interval we want, only the sine expansion on 0 ≤ x ≤ 1 would solve the problem. The ODE and BCs together picked the expansion and the interval. In fact, before the expansion was even constructed we had:
Which is a Fourier sine series only at t = 0. That the IC was defined at t = 0 allowed the expansion. For t > 0, the solution has nothing in common with a Fourier series.
What's trying to be emphasized is flexibility. Knowledge of Fourier series makes it much easier to solve problems. In the parallel plate problem, knowing what a Fourier sine series is motivates the construction of the sum of un. In the end it's the problem that dictates what needs to be done. For the separable IBVPs, expansions will be a recurring nightmare theme and it is most important to be familiar and comfortable with orthogonality and its application to making sense out of infinite sums. Many functions have orthogonality properties, including Bessel functions, Legendre polynomials, and others.
The keyword is orthogonality. If an orthogonality relation exists for a given situation, then a series solution is easily possible. As an example, the diffusion equation used in the previous chapter can, with sufficiently ugly BCs, require a trigonometric series solution that is not a Fourier series (non-integer, not evenly spaced frequencies of the sinusoids). Sturm-Liouville theory rescues us in such cases, providing the right orthogonality relation.
Numeric Methods
Finite Difference Method
The finite difference method is a basic numeric method which is based on the approximation of a derivative as a difference quotient. We all know that, by definition:
The basic idea is that if is "small", then
Similarly,
It's a step backwards from calculus. Instead of taking the limit and getting the exact rate of change, we approximate the derivative as a difference quotient. Generally, the "difference" showing up in the difference quotient (ie, the quantity in the numeriator) is called a finite difference which is a discrete analog of the derivative and approximates the derivative when divided by .
Replacing all of the derivatives in a differential equation ditches differentiation and results in algebraic equations, which may be coupled depending on how the discretization is applied.
For example, the equation
may be discretized into:
This discretization is nice because the "next" value (temporally) may be expressed in terms of "older" values at different positions.
Method of Lines
Introduction
The method of lines is an interesting numeric method for solving partial differential equations. The idea is to semi-discretize the PDE into a huge system of (continuous and interdependent) ODEs, which may in turn be solved using the methods you know and love, such as forward stepping Runge-Kutta. This is where the name "method of lines" comes from: the solution is composed of a juxtaposition of lines (curves, more properly).
The method of lines is applicable to IBVPs only. The variables (or variable) that have boundary constraints are discretized, and the (one) variable that is associated with the initial value(s) becomes the independent variable for the ODE solution. Put more formally, the method of lines is obtained when all of the variables are discretized except for one of them, resulting in a coupled system of ODEs.
Pure BVPs, such as Laplace's equation on some terrible boundary, can be solved in a manner similar to Jacobi iteration known as the method of false transients.
Example Problem
Consider the following nonlinear IBVP, where :
This intimidating nondimensional system describes the flow of liquid in an interior corner ("groove"), where is the height of the liquid and is the distance along the axis of the corner. The IBVP states that the corner is initially empty, and that at one end () the height is fixed to 1. Naturally, fluid will flow along , increasing until the other boundary (), where is always zero, is reached. It's worth noting that though this IBVP can't be solved analytically, similarity solutions to the nonlinear PDE exist for some other situations, such as fixed height at z = 0 and no constraint to the right of that.
The boundary values are associated with the spatial variable, . Suppose that is discretized in space but not time to points on a uniform grid. Then is approximated as follows:
So is a sequence (or vector) that has an index instead of a continuous dependence on ; note however that the whole sequence still has a continuous dependence on time. is an index that runs from 0 to , note that zero based indexing is being used, so that corresponds to and corresponds to . Looking at the boundaries, we know that:
What about the points of the sequence inside the boundaries? Initially (at ),
Suppose now that the right side of the PDE is discretized however you like, so that:
If, for example, central differences were used, one would eventually arrive at:
To construct the method of lines approximation for this problem, first is replaced with , and then the right side of the equation is replaced with its discretization (the exact equality is a slight abuse of notation):
Since depends continuously on time and nothing else, this differential equation becomes:
Putting everything together in one place, the solution to is the solution to the following IVP:
Solving this problem gives the approximation for . Note that if a second order forward stepping method is used, such as second order Runge Kutta, this solution method will have approximately the same accuracy as the Crank-Nicolson method, but without simultaneous solution of a mess of algebraic equations, which would make a nonlinear problem prohibitively difficult since a tidy matrix solution wouldn't work.
The method of lines is especially popular in electromagnetics, for example, using the Helmholtz equation to simulate the passage of light through a lens for better lens design; the reason for the popularity is that the system of ODEs can (in this case) be solved analytically, so that the accuracy is limited only by the spatial discretization.
A word of caution: an explicit forward stepping method of lines solution bears similarity to a forward stepping finite difference method; so there is no reason to believe that the method of lines doesn't suffer the same stability issues.
A Third Order TVD RK Scheme in C
What follows is an efficient TVD RK scheme implemented in C. Memory is automatically allocated per need (but it will just assume that there is no problem in memory allocation) and freed when n = 0. Note that the solution will not be TVD (total variation diminishing) unless the discretization of is TVD also.
This isn't strictly a method of lines solver, of course; it may find use whenever some large, interdependent vector ODE must be stepped forward.
Other Topics
Scale Analysis
In the chapter on nondimensionalization, variables (both independent and dependent) were nondimensionalized and at the same time scaled so that they ranged from something like to . "Something like to " is the mentality.
Scale analysis is a tool that uses nondimensionalization to:
Understand what's important in an equation and, more importantly, what's not.
Gain insight into the size of unknown variables, such as velocity or temperature, before (even without) actually solving.
Simplify solution process (nondimensional variables ranging for to are very amiable).
Reduce dependence of the solution on physical parameters.
Allow higher quality numeric solution since variables that are of the same range maintain accuracy better on a computer.
Scale analysis is very common sense driven and not too systematic. For this reason, and since it is somewhat unnatural and hard to describe without endless example, it may be difficult to learn.
Before going into the concept, we must discuss orders of magnitude.
Orders of Magnitude and Big O Notation
Suppose that there are two functions and . It is said (and notated) that:
Visualization of the limit superior and limit inferior of f(x) as x increases without bound.
It's worth understanding fully this possibly obscure definition. , short for limit superior, is similar to the "regular" limit, only it is the limit of the upper bound. This concept, alongside limit inferior, is illustrated at right. This intuitive analysis will have to suffice here, as the precise definition and details of these special limits are rather complicated.
As a further example, the limits of the cosine function as increases without bound are:
With this somewhat off topic technicality hopefully understood, the statement that:
Is saying that near , the order (or size, or magnitude) of is bounded by . It's saying that isn't crazily bigger then near , and this is precisely notated by saying that the limit superior is bounded (the "regular" limit wouldn't work since oscillations would ruin everything). The notation involving the big O is rather surprisingly called "big O notation", it's also known as Landau notation.
Take, for example, at different points:
In the first case, the term will easily dominate for large . Even if the coefficient on that term is very near zero, for large enough x that term will dominate. Hence, the function is of order for large .
In the second case, near the first two terms are limiting to zero while the constant term, , isn't changing at all. It is said to be of order , notated as order above. Why O(1) and not O(2)? Both are correct, but O(1) is preferred since it is simpler and more similar to .
This may put forth an interesting question: what would happen if the constant term was dropped? Both of the remaining terms would limit to zero. Since we are looking at x near zero and not at zero,
This is because as approaches zero, the quadratic term gets smaller much faster then the linear term. It would also be correct, though kind of useless, to call the quantity O(1). It would be incorrect to state that the quantity is of order zero since the limit would not exist, not under any circumstance.
As implied above, is by no means a unique function. All of the following statements are true, simply because the limit superior is bounded:
While technically correct, these are very misleading statements. Normally, the simplest, smallest magnitude function g(x) is selected.
Before ending the monotony, it should also be mentioned that it's not necessary for to be smaller then near , only the limit superior must exist. The following two statements are also true:
But again, these are misleading and it's most proper to state that:
A relatively simple concept has been beaten to death, to the point of being confusing. It'll be more clear in context, and it'll be used more in later chapters for different purposes.
Scale Analysis on a Two Term ODE
Previously, the following BVP was considered:
Wipe away any memory of solving this simple problem, the concepts of this chapter do not look at the actual solution. The variables are nondimensionalized by defining new variables:
So that is scaled by , and is scaled by an unknown scale . Now note that, thanks to the scaling:
These are both true near zero. will be O(1) (this is read "of order one") when its scale is properly chosen. Using the chain rule, the ODE was turned into the following:
Now, if both and are of order one, then it is reasonable to assume that, at least at some point in the domain of interest:
This is by no means guaranteed to be true, however it is reasonable.
To identify the velocity scale, we can set the derivative equal to one and solve. There is nothing "illegal" about purposely setting the derivative equal to one since all we need is some equation to specify an unknown constant, . There is much freedom in defining this scale, because what this constant is and how it's found has no effect on the validity of the solution of the BVP (as long as it's not something stupid like ).
Since:
It follows that:
This velocity scale may be thought of as a characteristic velocity. It's a number that shows us what to expect the velocity to be like. The velocity could actually be larger or smaller, but this gives a general idea. Furthermore, this scale tells us how chaging various physical parameters will affect the velocity; there are four of them summarized into one constant.
Compare this result to the coefficient (underlined) on the complete solution, with u dimensional and nondimensional:
They differ by a factor of , but they are of the same order of magnitude. So, indeed, characterizes the velocity.
Words like "reasonable" and "assume" were used a few times, words that would normally lead to the uglier word "approximate". Relax: the BVP itself hasn't been approximated or otherwise violated in any way. We just used scale analysis to pick a velocity scale that:
Turned the ODE into something very easy to look at:
Gained good insight into what kind of velocity the solution will produce without finding the actual solution.
Note that a zero pressure gradient can no longer show itself in the ODE. This is by no means a restriction, since a zero pressure gradient would result in a zero velocity scale which would unconditionally result in zero velocity.
Scale Analysis on a Three Term PDE
The last section was still more of nondimensionalization then it was scale analysis. To just begin getting deeper into the subject, we'll consider the pressure driven transient parallel plate IBVP, identical to the above only with a time component:
See the change of variables chapter to recall the origins of this problem. Scales are defined as follows:
Again, the scale on is picked to make it an order one quantity (based on the BCs), and the scales on and are just letters representing unknown quantities.
The chain rule has been used to define derivatives in terms of the new variables. Instead of taking this path, recall that, given variables and (for the sake of example) and their respective scales and :
So that makes things much easier. Performing the change of variables:
In the previous section, there was one unknown scale and one equation, so the unknown scale could be easily and uniquely isolated. Now, there are two unknown scales but only one equation (no, the BCs/IC will not help). What to do?
The physical meaning of scales may be taken into consideration. Ask: "What should the scales represent?"
There is no unique answer, but good answers for this problem are:
characterizes the steady state velocity.
characterizes the response time: the time to establish steady state.
Once again, these are picked (however, for this problem there really aren't any other choices). In order to determine the scales, the physics of each situation is considered. There may not be unique choices, but there are best choices, and these are the "correct" choices. An understanding of what each term in the PDE represents is vital to identifying these "correct" choices, and this is notated below:
For the velocity scale, a steady state condition is required. In that case, the time derivate (acceleration) must small. We could obtain the characteristic velocity associated with a steady state condition by requiring that the acceleration be something small (read: zero), stating that the second derivative is O(1), and solving:
This is the same as the velocity scale found in the previous section. This is expected since both situations are describing the same steady state condition. The neglect of acceleration equates to what's called a balance between driving force and viscosity since driving force and viscosity are all that remain.
Getting the time scale may be a little more elusive. The time associated with achieving steady state is dictated by the acceleration and the viscosity, so it follows that the time scale may be obtained by considering a balance between acceleration and viscosity. Note that this statement has nothing to do with pressure, so it should apply to a variety of disturbances. To balance the terms, pretend that the derivatives are O(1) quantities and disregard the pressure:
This is a statement that:
The smaller the viscosity, the longer you wait for steady state to be achieved.
The smaller the separation distance, the less you wait for steady state to be achieved.
Hence, the scale describes what will affect the transient time and how. The results may seem counterintuitive, but they are verified by experiment if the pressure is truly a constant capable of combating possibly huge viscosity forces for a high viscosity fluid.
Compare these scales to constants seen in the full, dimensional solution:
The velocity scales match in order of magnitude, nothing new there. But examine the time constant (extracted from the exponential factor) and compare to the time scale:
They are of the same order with respect to the physical parameters, though they'll differ by nearly a factor of 10 when n = 1. This result is more useful then it looks. Note that after determining the velocity scale, all three terms of the equation may have been considered to isolate a time scale. This would've been a poor choice that wouldn't have agreed with the time constant above since it wouldn't be describing the required settling between viscosity and acceleration.
Suppose that, for some problem, a time dependent PDE is too hard to solve, but the steady state version is easier and it is what you're interested in. A natural question would be: "How long do I wait until steady state is achieved?"
The time scale provided by a proper scale analysis will at least give an idea. In this case, assuming that the first term of the sum in the solution is dominant, the time scale will overestimate the response time by nearly a factor of 10, which is priceless information if you're otherwise clueless. This overestimate is actually a good (safe) overestimate, it's always better to wait longer and be certain of the steady state condition. Scales in general have a tendency to overestimate.
Before closing this section, consider the actual nondimensionalization of the PDE. During the scale analysis, the coefficients of the last two terms were equated and later the coefficients of the first two terms were equated. This implies that the nondimensionalized PDE will be:
And this may be verified by substituting the expressions found for the scales into the PDE. This dimensionless PDE, too, turned out to be completely independent of the physical parameters involved, which is very convenient.
Heat Flow Across a Thin Wall
Now, an important utility of scale analysis will be introduced: determining what's important in an equation and, better yet, what's not.
As mentioned in the introduction to the Laplacian, steady state heat flow in a homogeneous solid may be described by, in three dimensions:
Now, suppose we're interested in the heat transfer inside a large, relatively thin wall, with differing temperatures (not necessarily uniform) on different sides of the wall. The word 'thin' is crucial, write it down on your palm right now. You should suspect that if the wall is indeed thin, the analysis could be simplified somehow, and that's what we'll do.
Not caring about what happens at the edges of the wall, a BVP may be written:
is the thickness of the wall (implication: is the coordinate across the wall). Suppose that the wall is a boxy object with dimensions x x . Using the box dimensions as scales,
Only the scale of is unknown. Substituting into the PDE,
Note that the scale on divided out — so a logical choice must be made for it's scale; in this case it'd be an extreme boundary value (ie, the maximum value of ), let's say it's chosen and taken care of. Thanks to this scaling and the rearrangement that followed, we may get a good idea of the magnitude of each term in the equation:
Each derivative is approximately O(1). But what about the squared ratio of dimensions? This is called a dimensionless parameter. Look at your palm now (the one you don't write with), recall the word "thin". "Thin" in this case means exactly the same thing as:
And if the ratio above is much smaller then , then the square of this ratio is even smaller. Our dimensionless parameter is called a small parameter. When a parameter is small, there are many opportunities to simplify analysis; the simplest would be to state that it's too small to matter, so that:
What was just done couldn't have been justified without scaling variables so that their derivatives are (likely) O(1), since you have no idea what order they are otherwise. We know that each derivative is hopefully O(1), but some of these O(1) derivatives carry a very small factor. Only then can terms be righteously dropped. The dimensionless BVP becomes:
Note that it's still a partial differential equation (the and varialbes haven't been made irrelevant – look at the BCs). Also note that scaling on is undone since it cancels out anyway (the scale could've still been picked as, say, a maximum boundary value). This problem may be solved very simply by integrating the PDE twice with respect to , and then considering the BCs:
and are integration "constants". The first BC yields:
And the second:
The solution is:
It's just saying that the temperature varies linearly from one wall face to the other. It's worth noting that in practice, once scaling is complete, the hats on variables are "dropped" for neatness and to prevent carpal tunnel syndrome.
Words of Caution
Failure of one dimensional flow approximation.
"Extreme caution" is more fitting.
In the wall heat transfer problem, we took the partial derivatives in and to be O(1), and this was justified by the scaling: , and are O(1), so the derivatives must be so as well. Right?
Not necessarily. That they're O(1) is a linear approximation, however if the function is significantly nonlinear with respect to a variable of interest, then the derivatives may not be as O(1) as thought. In this problem, one way that this can happen is if the temperature at each wall face (the functions and ) have large and differing Laplacians. This will result in three dimensional heat conduction.
Examine carefully the image at right. Suppose that side length is ten times the wall thickness; and have zero Laplacians everywhere except along circles where temperatures suddenly change. At these locations, the Laplacian can be huge (unbounded if the sudden changes are discontinuities). This will suggest that the derivatives in question are not O(1) but much greater, so that these terms become important even though in this case:
Which is as required by the scale analysis: the wall is clearly thin. But apparently, the small thinness ratio multiplied by the large derivatives leads to significant quantities.
Both the exact solution and the solution to the problem approximated through scaling are shown at the location of a cutting plane. The exact solution shows at least two dimensional heat transfer, while the solution of the simplified solution shows only one dimensional heat transfer and is substantially different.
It's easy to see why the 1D approximation fails even without knowing what a Laplacian is: this is a heat transfer problem involving the diffusion of temperature, and the temperature will clearly need to diffuse along near the sudden changes within the wall (can't say the same about the BCs since they're fixed).
The caption of the figure starts with the word "failure". Is it really a failure? That depends on what you're looking for, it may or may not be. Note that if the wall were even thinner and the sudden jumps not discontinuities, the exact and 1D solutions could again eventually become indistinguishable.
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SMTE
1351:
Fundamentals of Math II
This course is the second
in a sequence of three mathematics content courses for students
seeking
certification for EC-6 Generalist, Special Education, Bilingual Education,
and grade 4-8 disciplines. This
course provides the conceptual framework for application of rational numbers, probability and statistics in
a problem solving setting.
"These courses should
be designed to ensure that the material is understood by the students
at a deeper level than would be the case if they took a more traditional
mathematics course....the material should be presented as much as possible
in a form that connects to the ways in which the subject comes up in
the elementary classroom....the courses should be such that they motivate
and engage students who have come to fear mathematics and mistrust their
own abilities to understand it al all. Finally, the course should involve
opportunities and requirements for communicating understanding of mathematics"
(Jonker, PRIMUS, 2008).
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Analogies 1 is a worktext designed to introduce students to analogy problem solving and to provide them with oppurtunities for vocabulary study. Its three-part organization is much like that of Analogies 2 and Analogies 3, the books which complete this series. However, as the introductory volume, Analogies 1 contains more basic vocabulary, and part 1, which presents analogies and strategies for solving them, contains more explanatory material and more exercises on this material. Both the content and the visual design of Part 1 will enlighten and reassure students who may be encountering analogies for the first time.
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Description
The Rockswold/Krieger algebra series uses relevant applications and visualization to show students why math matters, and gives them a conceptual understanding. It answers the common question "When will I ever use this?" Rockswold teaches students the math in context, rather than including the applications at the end of the presentation. By seamlessly integrating meaningful applications that include real data and supporting visuals (graphs, tables, charts, colors, and diagrams), students are able to see how math impacts their lives as they learn the concepts. The authors believe this approach deepens conceptual understanding and better prepares students for future math courses and life.
Table of Contents
1. Real Numbers and Algebra
1.1 Describing Data with Sets of Numbers
1.2 Operations on Real Numbers
1.3 Integer Exponents
1.4 Variables, Equations, and Formulas
1.5 Introduction to Graphing
Summary - Review Exercises - Test - Extended and Discovery Exercises
2. Linear Functions and Models
2.1 Functions and Their Representations
2.2 Linear Functions
2.3 The Slope of a Line
2.4 Equations of Lines and Linear Models
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-2 Cumulative Review Exercises
3. Linear Equations and Inequalities
3.1 Linear Equations
3.2 Introduction to Problem Solving
3.3 Linear Inequalities
3.4 Compound Inequalities
3.5 Absolute Value Equations and Inequalities
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-3 Cumulative Review Exercises
4. Systems of Linear Equations
4.1 Systems of Linear Equations in Two Variables
4.2 The Substitution and Elimination Methods
4.3 Systems of Linear Inequalities
4.4 Introduction to Linear Programming
4.5 Systems of Linear Equations in Three Variables
4.6 Matrix Solutions of Linear Systems
4.7 Determinants
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-4 Cumulative Review Exercises
5. Polynomial Expressions and Functions
5.1 Polynomial Functions
5.2 Multiplication of Polynomials
5.3 Factoring Polynomials
5.4 Factoring Trinomials
5.5 Special Types of Factoring
5.6 Summary of Factoring
5.7 Polynomial Equations
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-5 Cumulative Review Exercises
6. Rational Expressions and Functions
6.1 Introduction to Rational Functions and Equations
6.2 Multiplication and Division of Rational Expressions
6.3 Addition and Subtraction of Rational Expressions
6.4 Rational Equations
6.5 Complex Fractions
6.6 Modeling with Proportions and Variation
6.7 Division of Polynomials
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-6 Cumulative Review Exercises
7. Radical Expressions and Functions
7.1 Radical Expressions and Functions
7.2 Rational Exponents
7.3 Simplifying Radical Expressions
7.4 Operations on Radical Expressions
7.5 More Radical Functions
7.6 Equations Involving Radical Expressions
7.7 Complex Numbers
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-7 Cumulative Review Exercises
8. Quadratic Functions and Equations
8.1 Quadratic Functions and Their Graphs
8.2 Parabolas and Modeling
8.3 Quadratic Equations
8.4 The Quadratic Formula
8.5 Quadratic Inequalities
8.6 Equations in Quadratic Form
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-8 Cumulative Review Exercises
9. Exponential and Logarithmic Functions
9.1 Composite and Inverse Functions
9.2 Exponential Functions
9.3 Logarithmic Functions
9.4 Properties of Logarithms
9.5 Exponential and Logarithmic Equations
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-9 Cumulative Review Exercises
10. Conic Sections
10.1 Parabolas and Circles
10.2 Ellipses and Hyperbolas
10.3 Nonlinear Systems of Equations and Inequalities
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-10 Cumulative Review Exercises
11. Sequences and Series
11.1 Sequences
11.2 Arithmetic and Geometric Sequences
11.3 Series
11.4 The Binomial Theorem
Summary - Review Exercises - Test - Extended and Discovery Exercises
Chapters 1-11 Cumulative Review Exercises
Appendix A: Using the Graphing Calculator
Appendix B: The Midpoint Formula
Bibliography
Answers to Selected Exercises
Photo Credits
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The Accounting Foundations Workbook 3e has been revised to align with the New Zealand Curriculum 2007 and the realigned Achievement Standards for Level 1 of NCEA.
It contains hundreds of activities designed to help students understand the Accounting concepts covered for Level 1 of NCEA. Full answer...
Practice and consolidate skills
The author team of David Barton and Anna Cox has been joined by Philip Lloyd to provide a brand new, easy-to-use, write-on workbook. It contains approx 80 self-contained assignments linked to the Delta Mathematics textbook, making it easy for the teacher to give pra...
For Students
Geography 1.4 is a practical resource to help you pass the Geography 1.4 Achievement Standard: Apply concepts and basic geographic skills to demonstrate understanding of a given environment.
Features
A wide range of activities, covering:
Natural and cultural features; maps and m...
Geography 2.4 3rd edition covers all the skills (thinking, practical and valuing) and key geographic concepts needed to help Year 12 students prepare for Geography Achievement Standard 2.4 (Apply concepts and geographic skills to demonstrate understanding of a given environment), as well as 2.5 (Con...
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All Matters of Math by Canaa
Thanks for Visiting!
My name is Canaa Lee, and I am proud to announce that my dream of becoming an author has finally come true! My first book, Algebra for the Urban Student is the first algebra book
that was written for the typical math student. Math books are written for people who love and understand the jargon. From my experience as a high school teacher, most students dislike
mathematics because it has always been difficult for them and they have never been good at it.
Algebra for the Urban Student is a book that is written in the language that I use to personalize my classroom. I have interjected my personality in my book to guide my students
through their assignments in class. Now, when students leave my classroom, it is as if they have taken me home with them to help complete their assignments.
The chapters in Algebra for the Urban Student illustrate a significant algebra concept, such as solving linear equations and inequalities and finding the slope of a line. Then, the
chapter includes homework assignment that provides students with the opportunity to "demonstrate your understanding." In addition, there are real life projects for both algebra and geometry,
grading rubrics for whole and small class discussions. Furthermore, there are algebra 2 lessons that utilize the graphing calculator and takes the pain out of learning upper lever algebra.
This is just the first. I am writing the sequel to Algebra for the Urban Student. I anticipate its release in August 2012! You can purchase Algebra for the Urban
Student today!
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Looking for a software utility that would help math students plot various graphs? You've just stumbled upon the most hip one. GraphSight 2.0.1 is a newly updated popular and feature-rich comprehensive 2D graphing utility with easy navigation, perfectly suited for use by high-school and college math students. The program is capable of plotting Cartesian, polar, table defined, as well as specialty graphs, such as trigonometric functions (sin, cos, tg, and others). Importantly, it features a simple data and formula input format, making it very practical for solving in-class and homework algebra or calculus problems. The program comes with customizable axis options (color, style, width, grid), and table data import/export options. The one feature that makes it very popular among both math teachers and students is that the graphs the program plots are fully interactive. This object-oriented approach lets the students zoom in and out, see the points of intercept and do much more. Also, the program allows multiple graphs plotting. GraphSight is much easier to use than most similar software titles. Program's user interface is well labeled and is extremely simple to follow. The program runs under Windows 98/NT/ME/2000/XP and requires Internet Explorer 5.0 or higher installed. The price of a single copy is only 19 US Dollars. There is an unconditional 60-day money back guarantee. The benefits of registration include removal of all limitations, life-time worth of free updates and responsive technical support. Academic resellers, schools and colleges receive a significant discount for registering multiple copies. There also is a feature limited freeware version of the program called GraphSight Junior. Get your copy of GraphSight now!MatheGrafix - MatheGrafix is an excellent tool that allows you to plot 2D graphs including linear, geometric, exponential, fractals ones and more.MatheGrafix is an excellent tool that allows you to plot 2D graphs including linear, geometric, exponential,...GraphSight Match at Super Shareware
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MathCast's powerful and friendly graphical interface is suited for rapid development of mathematical equations. A part of this interface is called The Rapid Mathline. This method is intuitive and effective. The Rapid Mathline supports an extensive set of mathematical operators, symbols, and functions.
MathCast is also an Equation List Manager, and is capable of organizing dozens of equations in a single list. This empowers you with the abilities to manage, modify, view, edit, and reedit all the mathematics of a project (be it a document, a webpage, or so on) all at the same session.Equation Wizard - The program allows you to solve algebraic equations in the automatic mode.The program allows you to solve algebraic equations in the automatic mode. You just enter an equation in any form without any preparatory operations. Step by step EquationEquation - Equation is a useful software that can let you study and solve the equations of the second degree, enter only the values of a, b, and c and this software will do the rest.Equation is a useful software that can let you study and solve the equations
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Then one day in a store he saw a book titled "Calculus for the Practical Man" by Silvanus P. Thompson. Curious, and thinking himself quite practical, he opened the book and on the second page, all by itself, thus giving it great reverence, was the quote: "What one fool can do, another can. - Ancient Simian Proverb".As is true with any engineering major, I was required to take many math and computer programming classes, so you can be assured that I have extensive knowledge and practical understanding of these disciplines. Having coached my three high school students to As in math, I know I can help you sign...
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Elementary Number Theory with Applications
The advent of modern technology has brought a new dimension to the power of number theory: constant practical use. Once considered the purest of pure mathematics, it is used increasingly now in the rapid development of technology in a number of areas, such as art, coding theory, cryptology, computer science, and other necessities of modern life. Elementary Number Theory with Applications is the fruit of years of dreams and the author's fascination with the subject, encapsulating the beauty, elegance, historical development, and opportunities provided for experimentation and application. This is the only number theory book to show how modular systems can be employed to create beautiful designs, thus linking number theory with both geometry and art. It is also the only number theory book to deal with bar codes, Zip codes, International Standard Book Numbers (ISBN), and European Article Numbers (EAN). Emphasis is on problem-solving strategies (doing experiments, collecting and organizing data, recognizing patterns, and making conjectures). Each section provides a wealth of carefully prepared, well-graded examples and exercises to enhance the readers' understanding and problem-solving skills.
This is the only number theory book to: Show how modular systems can be employed to create beautiful designs, thus linking number theory with both geometry and art Deal with bar codes, Zip codes, International Standard Book Numbers (ISBN), and European Article Numbers (EAN) Emphasize problem-solving strategies (doing experiments, collecting and organizing data, recognizing patterns, and making conjectures) Provide a wealth of carefully prepared, well-graded examples and exercises to enhance the readers' understanding and problem-solving skills
This book is intended to serve as a one-semester introductory course in number theory and it includes a wealth of exercises. A historical perspective has been adopted and emphasis is given to some of ...
Group Theory is an indispensable mathematical tool in many branches of physics and chemistry. Providing a detailed account of Group Theory and its applications to chemical physics, the first half of ...
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Synopsis
Peterson's Master the SAT: Functions and Intermediate Algebra Review gives you the review and expert tips you need to help improve your score on the these types of questions on the Math part of the SAT. Here you can review functions, integer and rational expressions, solving complex equations, linear and quadratic functions, and more. In addition, the feature "Top 10 Strategies to Raise Your Score" offers expert tips to help you score high on rest of this important test. Master the SAT: Functions and Intermediate Algebra Review is part of Master the SAT 2011, which offers readers 6 full-length practice tests and in-depth review of the Critical Reading; Writing, and Math sections, as well as top test-taking tips to score high on the SAT
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0321652797
9780321652799
Using and Understanding Mathematics: Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases readers' mathematical literacy so that they better understand the mathematics used in their daily lives, and can use math effectively to make better decisions every day. Contents are organized with that in mind, with engaging coverage in sections like Taking Control of Your Finances, Dividing the Political Pie, and a full chapter about Mathematics and the Arts. Note: This is the standalone book, if you want the book with the Access Card please order the ISBN below: 0321727746 / 9780321727749 Using and Understanding Mathematics: A Quantitative Reasoning Approach with MathXL (12-month access) * Package consists of 0201716305 / 9780201716306 MathXL -- Valuepack Access Card (12-month access) 0321652797 / 9780321652799 Using and Understanding Mathematics: A Quantitative Reasoning Approach «Show less
Using and Understanding Mathematics: Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases readers' mathematical literacy so that they better understand the mathematics used in their daily lives, and can use math effectively to make... Show more»
Preface
xi
Acknowledgments
xviii
Prologue: Literacy for the Modern World
PART ONE Logic and Problem Solving
Thinking Critically
Activity Bursting Bubble
Recognizing Fallacies
Propositions and Truth Values
Sets and Venn Diagrams
A Brief Review: Sets of Numbers
Analyzing Arguments
Critical Thinking in Everyday Life
Approaches to Problem Solving
Activity Global Melting
The Problem-Solving Power of Units
A Brief Review: Working with Fractions
Using Technology: Currency Exchange Rates
Standardized Units: More Problem-Solving Power
A Brief Review: Powers of 10
Using Technology: Metric Conversions
Problem-Solving Guidelines and Hints
PART TWO Quantitative Information in Everyday Life
Numbers in the Real World
Activity Big Numbers
Uses and Abuses of Percentages
A Brief Review: Percentages
A Brief Review: What Is a Ratio?
Putting Numbers in Perspective
A Brief Review: Working with Scientific Notation
Using Technology: Scientific Notation
Dealing with Uncertainty
A Brief Review: Rounding
Using Technology: Rounding in Excel
Index Numbers: The CP1 and Beyond
Using Technology: The Inflation Calculator
How Numbers Deceive: Polygraphs, Mammograms, and More
Managing Money
Activity Student Loans
Taking Control of Your Finances
The Power of Compounding
A Brief Review: Powers and Roots
Using Technology: Powers
Using Technology: The Compound Interest Formula
Using Technology: The Compound Interest Formula for Interest Paid More than Once a Year
Using Technology: APY in Excel
Using Technology: Powers of e
A Brief Review: Three Basic Rules of Algebra
Savings Plans and Investments
Using Technology: The Savings Plan Formula
A Brief Review: Algebra with Powers and Roots
Using Technology: Fractional Powers (Roots)
Loan Payments, Credit Cards, and Mortgages
Using Technology: The Loan Payment Formula (installment Loans)
Using Technology: Principal and Interest Payments
Income Taxes
Understanding the Federal Budget
PART THREE Probability and Statistics
Statistical Reasoning
Activity Cell Phones and Driving
Fundamentals of Statistics
Using Technology: Random Numbers
Should You Believe a Statistical Study?
Statistical Tables and Graphs
Using Technology: Frequency Tables in Excel
Using Technology: Bar Graphs and Pie Charts in Excel
Using Technology: Fine Charts and Histograms in Excel
Graphics in the Media
Correlation and Causality
Using Technology: Scatter Diagrams in Excel
Putting Statistics to Work
Activity Bankrupting the Auto Companies
Characterizing Data
Using Technology: Mean, Median, Mode in Excel
Measures of Variation
Using Technology: Standard Deviation in Excel
The Normal Distribution
Using Technology: Standard Scores in Excel
Using Technology: Normal Distribution Percentiles in Excel
Statistical Inference
Probability: Living with the Odds
Activity Lotteries
Fundamentals of Probability
A Brief Review: The Multiplication Principle
Combining Probabilities
The Law of Large Numbers
Assessing Risk
Counting and Probability
A Brief Review: Factorials
Using Technology: Factorials
Using Technology: Permutations
Using Technology: Combinations
PART FOUR Modeling
Exponential Astonishment
Activity Towers of Hanoi
Growth: Linear versus Exponential
Doubling Time and Half-Life
A Brief Review: Logarithms
Using Technology: Logarithms
Real Population Growth
Logarithmic Scales: Earthquakes, Sounds, and Acids
Modeling Our World
Activity Bald Eagle Recovery
Functions: The Building Blocks of Mathematical Models
A Brief Review: The Coordinate Plane
Linear Modeling
Using Technology: Graphing Functions
Exponential Modeling
A Brief Review: Algebra with Logarithms
Modeling with Geometry
Activity Eyes in the Sky
Fundamentals of Geometry
Problem Solving with Geometry
Fractal Geometry
PART FIVE Further Applications
Mathematics and the Arts
Activity Digital Music Files
Mathematics and Music
Perspective and Symmetry
Proportion and the Golden Ratio
Mathematics and Politics
625
Activity Congressional District Boundaries
Voting: Does the Majority Always Rule?
Theory of Voting
Apportionment: The House of Representatives and Beyond
Dividing the Political Pie
665
Credits
Answers
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This volume contains the basics of what every scientist and engineer should know about complex analysis. A lively style combined with a simple, direct approach helps readers grasp the fundamentals, from complex numbers, limits in the complex plane, and complex functions to Cauchy's theory, power series, and applications of residues. 1974 edition. Reprint of the Prentice-Hall, 1974Counterexamples in Analysis by Bernard R. Gelbaum
John M. H. Olmsted These counterexamples deal mostly with the part of analysis known as "real variables." Covers the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, functions of 2 variables, plane sets, more. 1962 edition. read more
$15.95
Foundations of Analysis: Second Edition by David F Belding
Kevin J Mitchell Unified and highly readable, this introductory approach develops the real number system and the theory of calculus, extending its discussion of the theory to real and complex planes. 1991 edition. read more
$24.95
A Second Course in Complex Analysis by William A. Veech Geared toward upper-level undergraduates and graduate students, this clear, self-contained treatment of important areas in complex analysis is chiefly classical in content and emphasizes geometry of complex mappings. 1967 editionAsymptotic Methods in Analysis by N. G. de Bruijn This pioneering study/textbook in a crucial area of pure and applied mathematics features worked examples instead of the formulation of general theorems. Extensive coverage of saddle-point method, iteration, and more. 1958 edition. read more
Applied Analysis by Cornelius Lanczos Classic work on analysis and design of finite processes for approximating solutions of analytical problems. Features algebraic equations, matrices, harmonic analysis, quadrature methods, and much more. read more
$24.95
Foundations of Modern Analysis by Avner Friedman Measure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces — all in a single study. Detailed analyses. Problems. Bibliography. Index. read more
$11.95
Foundations of Mathematical Analysis by Richard Johnsonbaugh
W.E. Pfaffenberger Definitive look at modern analysis, with views of applications to statistics, numerical analysis, Fourier series, differential equations, mathematical analysis, and functional analysis. More than 750 exercises; some hints and solutions. 1981 edition. read more
$22
$17.95
Topology for Analysis by Albert Wilansky Three levels of examples and problems make this volume appropriate for students and professionals. Abundant exercises, ordered and numbered by degree of difficulty, illustrate important topological concepts. 1970 edition. read more
$22.95
Intermediate Mathematical Analysis by Anthony E. Labarre, Jr. Focusing on concepts rather than techniques, this text deals primarily with real-valued functions of a real variable. Complex numbers appear only in supplements and the last two chapters. 1968 edition. read more
$15.95
An Introduction to Mathematical Analysis by Robert A. Rankin Dealing chiefly with functions of a single real variable, this text by a distinguished educator introduces limits, continuity, differentiability, integration, convergence of infinite series, double series, and infinite products. 1963 edition. read more
Analysis in Euclidean Space by Kenneth Hoffman Developed for a beginning course in mathematical analysis, this text focuses on concepts, principles, and methods, offering introductions to real and complex analysis and complex function theory. 1975 edition. read more
Real Analysis by Gabriel Klambauer Concise in treatment and comprehensive in scope, this text for graduate students introduces contemporary real analysis with a particular emphasis on integration theory. Includes exercises. 1973 edition. read more
$22.95
Applied Nonstandard Analysis by Prof. Martin Davis This applications-oriented text assumes no knowledge of mathematical logic in its development of nonstandard analysis techniques and their applications to elementary real analysis and topological and Hilbert space. 1977 edition. read more
$14.95
Introduction to Analysis by Maxwell Rosenlicht Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition. read more
$14.95
Introductory Complex Analysis by Richard A. Silverman Shorter version of Markushevich's Theory of Functions of a Complex Variable, appropriate for advanced undergraduate and graduate courses in complex analysis. More than 300 problems, some with hints and answers. 1967 edition. read more
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Mathematicians
Onet SOC Code 15-2021.00
Career Description:
Conduct research in fundamental mathematics or in application of mathematical techniques to science, management, and other fields. Solve or direct solutions to problems in various fields by mathematical methods.
Related Clusters:
Programs of Study
Career Summary:
Job Zone Five: Extensive Preparation Needed
Experience-- Training- Employees may need some on-the-job training, but most of these occupations assume that the person will already have the required skills, knowledge, work-related experience, and/or training.
Examples-Physics - Knowledge and prediction of physical principles, laws, their interrelationships, and applications to understanding fluid, material, and atmospheric dynamics, and mechanical, electrical, atomic and sub- atomic structures and processes.
Abilities
Mathematical Reasoning - The ability to choose the right mathematical methods or formulas to solve a problem.
Oral Comprehension - The ability to listen to and understand information and ideas presented through spoken words and sentences.
Written Comprehension - The ability to read and understand information and ideas presented in writing.
Originality - The ability to come up with unusual or clever ideas about a given topic or situation, or to develop creative ways to solve a problem.
Number Facility - The ability to add, subtract, multiply, or divide quickly and correctly.
Fluency of Ideas - The ability to come up with a number of ideas about a topic (the number of ideas is important, not their quality, correctness, or creativity).
Deductive Reasoning - The ability to apply general rules to specific problems to produce answers that make sense.
Oral Expression - The ability to communicate information and ideas in speaking so others will understand.
Inductive Reasoning - The ability to combine pieces of information to form general rules or conclusions (includes finding a relationship among seemingly unrelated eventsSpeed of Closure - The ability to quickly make sense of, combine, and organize information into meaningful patterns.
Category Flexibility - The ability to generate or use different sets of rules for combining or grouping things in different ways.
Near Vision - The ability to see details at close range (within a few feet of the observer).
Flexibility of Closure - The ability to identify or detect a known pattern (a figure, object, word, or sound) that is hidden in other distracting material.
Problem Sensitivity - The ability to tell when something is wrong or is likely to go wrong. It does not involve solving the problem, only recognizing there is a problem.
Making Decisions and Solving Problems - Analyzing information and evaluating results to choose the best solution and solve problems.
Interacting With Computers - Using computers and computer systems (including hardware and software) to program, write software, set up functions, enter data, or process information.
Interpreting the Meaning of Information for Others - Translating or explaining what information means and how it can be used.
Provide Consultation and Advice to Others - Providing guidance and expert advice to management or other groups on technical, systems-, or process-related topics.
Communicating with Supervisors, Peers, or Subordinates - Providing information to supervisors, co-workers, and subordinates by telephone, in written form, e-mail, or in person - Estimating sizes, distances, and quantities; or determining time, costs, resources, or materials needed to perform a work activity.
Interests
Investigative - Investigative occupations frequently involve working with ideas, and require an extensive amount of thinking. These occupations can involve searching for facts and figuring out problems mentallyAutonomy - Workers on this job plan their work with little supervision.
Ability Utilization - Workers on this job make use of their individual abilitiesIndependence - Workers on this job do their work alone.
Working Conditions - Workers on this job have good working conditions.
Responsibility - Workers on this job make decisions on their own.
Achievement - Workers on this job get a feeling of accomplishment.
Security - Workers on this job have steady employment.
Creativity - Workers on this job try out their own ideas.
Moral Values - Workers on this job are never pressured to do things that go against their sense of right and wrong.
Working Conditions-Mean Extent - Occupations that satisfy this work value offer job security and good working conditions. Corresponding needs are Activity, Compensation, Independence, Security, Variety and Working Conditions.
Activity - Workers on this job are busy all the time.
Social Status - Workers on this job are looked up to by others in their company and their community.
Company Policies and Practices - Workers on this job are treated fairly by the company.
Recognition - Workers on this job receive recognition for the work they do.
Compensation - Workers on this job are paid well in comparison with other workers.
Recognition-Mean Extent - Occupations that satisfy this work value offer advancement, potential for leadership, and are often considered prestigious. Corresponding needs are Advancement, Authority, Recognition and Social Status.
Tennessee Board of Regents is an AA/EEO employer and does not discriminate on the basis of race, color, national origin, sex, disability, or age in its programs and activities. Full Non-Discrimation Policy.
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Mathematics
Pupils who decide to study sixth form mathematics should be prepared for a very exciting, nerve-wracking, fretful, fun-filled, occasionally frustrating, but ultimately rewarding couple of years. At least, that is what we aim for in the King's Mathematics Department.
AS and A2 mathematics and further mathematics are not easy courses, in spite of what some newspapers, and talking heads seem to indicate. Thus, for a student to be successful they will require, most obviously and importantly, an interest and enjoyment in the subject. This, allied with superior algebraic skills, should allow the student to emerge triumphant at the end of their course.
Sixth form mathematics is taught by experienced teachers in a generally relaxed setting, where students will be expected to contribute to their own learning, both in the classroom and via independent study.
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The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.
A Guide to Complex Variables gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource. In addition to important ideas from the Cauchy theory, the book also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics. Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alikeSink or Float: Thought Problems in Math and Physics is a collection of problems drawn from mathematics and the real world. Its multiple-choice format forces the reader to become actively involved in deciding upon the answer. The book's aim is to show just how much can be learned by using everyday common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem's solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or little-known facts. The problems themselves can easily turn into serious debate-starters, and the book will find a natural home in the classroom.
Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often, especially in secondary and collegiate mathematics, the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don't possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how the use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and the authors will convince you that the same is true when working with inequalities. They show how to produce figures in a systematic way for the illustration of inequalities and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument cannot only show two things unequal, but also help the observer see just how unequal they are. The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.
A Guide to Advanced Real Analysis is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.
A Guide to Real Variables provides aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated. The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject resourceA Guide to Topology is an introduction to basic topology. It covers point-set topology as well as Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations researchCHOICE Award winner! A Guide to Elementary Number Theory is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams. Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series.
Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors' previous books with the MAA (Math Made Visual and When Less Is More), secondary school and college and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving. sports. The section on football includes an article that evaluates a method for reducing the advantage of the winner of a coin flip in an NFL overtime game; the section on track and field examines the ultimate limit on how fast a human can run 100 meters; the section on baseball includes an article on the likelihood of streaks; the section on golf has an article that describes the double-pendulum model of a golf swing, and an article on modeling Tiger Wood's career. The articles provide source material for classroom use and student projects. Many students will find mathematical ideas motivated by examples taken from sports more interesting than the examples selected from traditional sources.
Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.
Icons of mathematics are certain geometric diagrams that play a crucial role in visualizing mathematical proofs, and in the book the authors present 20 of them and explore the mathematics that lies within and that can be created. The authors devote a chapter to each icon, illustrating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathematical facts. Among these are classical results from plane geometry, properties of the integers, means and inequalities, trigonometric identities, theorems from calculus, and puzzles from recreational mathematics.
This book can be used in a one semester undergraduate course or senior capstone course, or as a useful companion in studying algebraic geometry at the graduate level. This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included. Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles' proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.
This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.
The purpose of A Guide to Functional Analysis is to introduce the reader with minimal background to the basic scripture of functional analysis. Readers should know some real analysis and some linear algebra. Measure theory rears its ugly head in some of the examples and also in the treatment of spectral theory. The latter is unavoidable and the former allows us to present a rich variety of examples. The nervous reader may safely skip any of the measure theory and still derive a lot from the rest of the book. Apart from this caveat, the book is almost completely self-contained; in a few instances we mention easily accessible references. A feature that sets this book apart from most other functional analysis texts is that it has a lot of examples and a lot of applications. This helps to make the material more concrete, and relates it to ideas that the reader has already seen. It also makes the book more accessible to a broader audience.
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0198507631
9780198507635
Oxford Users' Guide to Mathematics: The Oxford Users' Guide to Mathematics represents a comprehensive handbook on mathematics. It covers a broad spectrum of mathematics including analysis, algebra, geometry, foundations of mathematics, calculus of variations and optimization, theory of probability and mathematical statistics, numerical mathematics and scientific computing, and history of mathematics. This is supplemented by numerous tables on infinite series, special functions, integrals, integral transformations, mathematical statistics, and fundamental constants in physics. The book offers a broad modern picture of mathematics starting from basic material up to more advanced topics. It emphasizes the relations between the different branches of mathematics and the applications of mathematics in engineering and the natural sciences. The book addresses students in engineering, mathematics, computer science, natural sciences, high-school teachers, as well as a broad spectrum of practitioners in industry and professional researchers. A comprehensive table at the end of the handbook embeds the history of mathematics into the history of human culture. The bibliography represents a comprehensive collection of the contemporary standard literature in the main fields of mathematics. «Show less
Oxford Users' Guide to Mathematics: The Oxford Users' Guide to Mathematics represents a comprehensive handbook on mathematics. It covers a broad spectrum of mathematics including analysis, algebra, geometry, foundations of mathematics, calculus of variations and optimization,... Show more»
Rent Oxford Users' Guide to Mathematics today, or search our site for other Zeid
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Abstract Algebra (Hardcover) physicist......more physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.
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To reinforce the concepts of calculus like the connectivity of a behavior, the rate of change of a phenomena, extrema of a relation, composition and inverse of multiple operations, the concept of convergence and clustering in sequential steps, the concept of integration as accumulation with its vast applicability, etc.. By inducing further experimental formation, we will have satisfied the three basic objectives of calculus reform sought worldwide, namely the theoretical, the graphical and the visual aspect of such a rich subject. By completing the course, the student will have acquired a professional skill in handling phenomenal variations. It will enable him to master the vocabulary of the subject by grasping the fundamental concepts treated in calculus and treated by calculus.
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Nine PlanetsA Multimedia Tour of the Solar System: one star, eight planets, and more
Search for Adelphi, MD PhysicsAlgebra is generally the introduction to higher mathematics. While the ability to manipulate equations in order to solve problems is essential, student will also use verbal, numerical, graphic, and symbolic forms in the course. Algebra 2 is a continuation of topics first discussed in Algebra 1.
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Mathematics Club: Math Factor
Math Factor is a student organization that involves all students who's interest in mathematics. Its goal is to bring together people with love for mathematics, encourage student collaboration, provide exposure to further education and career opportunity. Math Factor holds academic and social events throughout the year. Among its activities are undergraduate research seminar, Integration bee, Pi day celebration, movie night, Bowling for Pi, etc.
Math Factor is a place for students interested in mathematics can get together and further explore mathematics outside of a classroom with various activities.
Scholarships
The Mathematics Department awards the following scholarships to students majoring in mathematics, as funding permits. If you wish to be considered for one of these scholarships, please fill out an information sheet, available in the Mathematics Department office. For deadline and detailed information, please contact Mathematics Department.
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Product Description
This program is going to hit you with some new words and definition, so get ready and take your position as we tackle key concepts and terms of algebra!Topics Covered: Monomials and Polynomials Simplifying Monomials Greatest Common Factor (GCF) Factoring and Simplifying Rational ExpressionsIncludes a DVD plus a CD-ROM with teacher's guide, quizzes, graphic organizers and classroom activities. Teaching Systems programs are optimized for classroom use and include "Full Public Performance Rights".Grade Level: 8-12
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Making Connections: Functions from Dot Products
Calculus is often a prerequisite course to linear algebra. As such, it is beneficial for students to see connections between concepts from linear algebra and more familiar topics from Calculus. This demo allows the instructor to tie together the concepts of dot product and a function—we use dot products to define a function that can be graphed in the plane. The ideas of orthogonality and parallel vectors can then be connected to the concepts from calculus that are important in the study of curve sketching.
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Resources
Here are some
lecture notes for Introduction to Algebra, a prerequisite for this course. If
you feel unsure about proof by induction, the definition of a group, or
the relation of congruence, you may find some help in these notes.
Here is some advice about study (lectures,
coursework, classes, and exams).
Here are links to some Theorems of the Day on
number theory. You may have seen some of these on the screens in the
Mathematics building. You might also like to read about the Congruent Numbers
problem here.
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Aims: To consolidate and extend topics met at A-level.
To improve students' fluency and understanding of the basic techniques required for engineering analysis.
Learning Outcomes: After taking this unit the student should be able to:
Handle circular and hyperbolic functions, and sketch curves. Differentiate and integrate elementary functions, products of functions etc.
Use complex numbers.
Employ standard vector and matrix techniques for geometrical purposes. Determine the Fourier series of a periodic function.
Understand power series representations of functions and their convergence properties.
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Neptune PrecalculusSolutions to problems and analysis of topics are mastered by applying logic, step by step processing, and introducing strategies that can assist quickly when used repetitively. I believe in developing a deeper understanding of the topic through comparisons and applications, and presenting the leAlso
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You will solve equations, graph, use formulas, etc. If you are somewhat nervous about this course, that is normal. But don't worry, we will get through this together. I will do my part to help you achieve your goals by teaching you, encouraging you, and getting you on the right course. My goal is for all of my students to pass this course but you must also do your part. These are the things you need to do:
1.Come to class everyday (except in case of an emergency, of course).
2.Come prepared. (Bring notebook, pencil, etc.)
3.Write the bell ringers down (mathstarter/warmup) and try to solve the problems. This is especially important for the 9th and 10th graders. The bell ringers help prepare you for the iLeap and GEE test.
4.Take notes and Ask Questions!!! This is where a lot of students mess up. I know it requires work - paying attention, using your hands to write, ... but I have confidence that you can do it. Anything I write on the board, you need to write down. If you don't have notes, what will you use to study or help you do your homework? Do not be afraid to ask questions. I will be more than happy to answer any question you have pertaining to the lesson.
5.Do your homework. Doing your homework lets you know if you understand a concept. It also prepares you for your test.
6.If you need more help, go to the websites on my resource page and come to tutoring.
If you do All of the Above, I truly believe that you can pass this course. It's your choice, so make the right one.
class material
Supply List
The following materials are needed and should be brought to class everyday.
1 to 1 1/2 in binder
pack of dividers
looseleaf paper
pencils
red pen or color pen, excluding black or blue
two(2) dry erase marker(preferably expo black)
calculator ( Scientific)
Things to know:
Each student is expected to maintain a binder that will be graded regularly.
There will be a home assignment daily. If problems are assigned; all work must be shown.
Students will have 1 to 2 test or quizzes per week.
Unit assessments will be standardized and created by the Parish. It will follow the High School Comprehensive Curriculum.
Grades will be determined based on total points that your child earns during the grading period. The school grading scale is then used to determine the letter grade.
Class Schedule
1st period Duty
2nd period Algebra I
3rd period Algebra I
4th period Algebra I
5th period Algebra I
6th period Planning Period
7th period Algebra I
Attendance
Algebra I Attendance:
To achieve success in Mathematics, it is important for students to attend class regularly. Many students have great difficulty trying to learn Algebra work on their own after an absence. Class work tends to build on prior learning. If a student misses just one day, they have difficulty following new assignments. Attendance is taken at every class period. Students must bring a note from a parent for each day missed which explains the reason for the absence. Students have two days following an absence to make arrangements with a teacher to make up work. Most often, test make-ups are given after school and at the teacher's convenience.
East Baton Rouge Parish School System is an equal opportunity employer and does not discriminate on the basis of race, color, national origin, gender, age, or qualified disability.
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Computer-aided assessment of maths, stats and numeracy from GCSE to undergraduate level 2. These resources have been made available under a Creative Common licence by Martin Greenhow and Abdulrahman Kamavi, Brunel University.
Support material from the University of Plymouth:
The output from this project is a library of portable, interactive, web based support packages to help students learn various mathematical ideas and techniques and to support classroom teaching.
There are support materials on ALGEBRA, GRAPHS, CALCULUS, and much more.
This material is offered through the mathcentre site courtesy of Dr Martin Lavelle and Dr Robin Horan from the University of Plymouth.
In this unit we see how the three trigonometric ratios cosecant, secant and cotangent can appear in trigonometric identities and in the solution of trigonometric equations. Graphs of the functions are obtained from a knowledge of sine, cosine and tangent. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
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Mathematics, BA
What Is the Study of Mathematics?
Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
As a practical matter, Mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, Mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
-From Everybody Counts: A Report to the Nation on the Future of Mathematics Education (c) 1989 National Academy of Sciences
Why Should I Consider this Major?
The special role of Mathematics in education is a consequence of its universal applicability. The results of Mathematics-theorems and theories-are both significant and useful; the best results are also elegant and deep. Through its theorems, Mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, Mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power-a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live.
-From Everybody Counts: A Report to the Nation on the Future of Mathematics Education (c) 1989 National Academy of Sciences
Empowered with the critical thinking skills that Mathematics develops, recent Mathematics graduates from Western have obtained positions in a variety of fields including actuarial science, cancer research, computer software development, business management and the movie industry. The skills acquired in our program have prepared graduates for further academic studies in Mathematics, Computer Science, Physics, Biology, Chemistry, Oceanography and Education.
Compared to the BS Mathematics major, the BA has fewer advanced requirements and provides greater flexibility in the choice of courses to take.
How to Declare:
Students who intend to complete a major in Mathematics are urged to declare the major formally at an early point in their Western career so that a program of study can be planned carefully in collaboration with a departmental advisor.
Coursework
MATH 204 Elementary Linear Algebra
MATH 224 Multivariable Calculus and Geometry I
MATH 226 Limits and Infinite Series
MATH 331 Ordinary Differential Equations
Choose either:
MATH 124 Calculus and Analytic Geometry I
MATH 125 Calculus and Analytic Geometry II
or
MATH 134 Calculus I Honors
MATH 135 Calculus II Honors
or
MATH 138 Accelerated Calculus
One course from:
MATH 341 Probability and Statistical Inference
MATH 441 Probability
One course from:
MATH 419 Historical Perspectives of Mathematics
MATH 420 Topics in the History and Philosophy of Mathematics
Note: The pair MATH 203 and 303 may be substituted for MATH 204 and 331
One course from:
CSCI 139 Programming Fundamentals in Python
CSCI 140 Programming Fundamentals in C++
CSCI 141 Computer Programming I
MATH 307 Mathematical Computing
Note: If the supporting sequence from CSCI below is chosen, this requirement is fulfilled.
Three courses from:
MATH 302 Introduction to Proofs Via Number Theory
MATH 304 Linear Algebra
MATH 309 Introduction to Proof in Discrete Mathematics
MATH 312 Proofs in Elementary Analysis
MATH 360 Euclidean and Non-Euclidean Geometry
Two courses from:
MATH 410 Mathematical Modeling
M/CS 335 Linear Optimization
M/CS 375 Numerical Computation
M/CS 435 Nonlinear Optimization
M/CS 475 Numerical Analysis
16 approved credits in mathematics or math-computer science, which includes completion of two of the following pairs:
One course from:
MATH 303 Linear Algebra and Differential Equations II
MATH 331 Ordinary Differential Equations
Together with one of:
MATH 415 Mathematical Biology
MATH 430 Fourier Series & Apps. to Partial Differential Equations
MATH 431 Analysis of Partial Differential Equations
MATH 432 Systems of Differential Equations
Only one of the pairs from the above group can be used
The following pair:
MATH 341 Probability and Statistical Inference
MATH 342 Statistical Methods
The following pair:
MATH 401 Introduction to Abstract Algebra
MATH 402 Introduction to Abstract Algebra
The following pair
MATH 421 Methods of Mathematical Analysis I
MATH 422 Methods of Mathematical Analysis II
The following pair
MATH 441 Probability
MATH 442 Mathematical Statistics
The following pair
M/CS 335 Linear Optimization
M/CS 435 Nonlinear Optimization
The following pair
M/CS 375 Numerical Computation
M/CS 475 Numerical Analysis
Note: Courses counted toward the major in the preceding boxes do not count toward the 16 credits but can serve as part (or all) of the pair.
One of the following sequences:
PHYS 121 Physics With Calculus I
PHYS 122 Physics With Calculus II
PHYS 123 Electricity and Magnetism
or
CHEM 121 General Chemistry I
CHEM 122 General Chemistry II
CHEM 123 General Chemistry III
or
CHEM 125 General Chemistry I, Honors
CHEM 126 General Chemistry II, Honors
CHEM 225 General Chemistry III, Honors
or
CSCI 141Computer Programming 1
CSCI 145 Computer Programming and Linear Data Structures
CSCI 241 Data Structures
CSCI 301 Formal Languages and Functional Programming
And one of:
CSCI 305 Analysis of Algorithms and Data Structures I
CSCI 330 Database System
CSCI 345 Object Oriented Design
CSCI 401 Automata and Formal Language Theory
ECON 206 Introduction to Microeconomics
ECON 207 Introduction to Macroeconomics
ECON 306 Intermediate Microeconomics
and one of
ECON 375 Introduction to Econometrics
ECON 470 Economic Fluctuations and Forecasting
ECON 475 Econometrics
GURs:
The courses below satisfy GUR requirements and may also be used to fulfill major requirements.
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Mathematics is one of the complex subjects. The students have to make a lot of
efforts to learn mathematics. To improve their skill in mathematics the students
have to join private institutes and pay ...
A function is that mathematical term which states relationship between constants
and one or more variables.
For example, consider a function f(x) = 7x4 + 100, which expresses a relationship
between ...
Graphing exponential functions is similar to the graphing you have done before.
However, by the nature of exponential functions, their points tend either to be
very close to one fixed value or else ...
Rational and Irrational Numbers :- Numbers appear like dancing letters to
many students as they are not able to distinguish between different categories of
numbers and get confused in understanding ...
Right Triangle :- A right triangle (American English) or right-angled triangle
(British English) is a triangle in which one angle is a right angle (that is, a 90-
degree angle). The relation between ...
Curve Fitting :- Curve fitting is the process of constructing a curve, or
mathematical function, that has the best fit to a series of data points, possibly
subject to constraints. Curve fitting can ...
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An exploration of the key issues in the teaching of mathematics, a key subject in its own right, and one that forms an important part of many other disciplines. The volume includes contributions from a wide range of experts in the field, and has a broad and international perspective. It is part of a series on effective learning and teaching in higher education. Each volume in the series contains advice, guidance and expert opinion on teaching in the key subjects in higher education today, and are backed up by the authority of the Institute for Learning and Teaching.
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Recommendation: Recommended. Description: User works through the line thinning algorithm to demonstrate proficiencyEvaluation: Surprisingly, its not the exercise that makes this AV, even though the exercise aspect is usually the strong point for Trakla AVs. In this case, stepping through the 'model answer' for an example does a really nice job of getting the algorithm across. Unfortunately, the exercise itself is hard to figure out how to get the mechanics right.
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Many calculators have features that are suitable to handle Algebra, mathematical graphs, problem solving and Calculus. Some include features that are unique to engineering students. Very few calculators reach out to students studying science. That is disappointing because many students have to purchase multiple calculators to use in different classes.
The Texas Instruments TI-83-Plus Silver Edition contains features that useful for students studying algebra, calculus, statistics, biology, chemistry, or physics. The TI-83-Plus calculator is the one and only calculator that students will need to meet their requirements while in school.
Features
·TI-Graph Link for Windows and unit-to-unit cable
·1.5MB of memory
·Store up to 94 applications
·Split screen to display graph and editor or graph and table
·Pre-loaded with popular Handheld Software Applications
·Interactive equation solver
The first thing you will notice about the TI-83-Plus calculator is it sleek design and appearance. Most other calculators carry a very staid black or dark color appearance. The TI-83-Plus stands out from the crowd with the silver casing.
Once you turn on the TI-83-Plus calculator you will be amazed at the wide range of features that it has. It can easily handle a matrix, graphing, or equation solver. For science students there is a pre-programmed periodic table. The mathematic tools are substantial, including viewing equations, graphs, and coordinate simultaneously; advanced statistics and regression analysis, graphical analysis, and data analysis; and tools for engineering, financial, logarithm, trigonometry, and hyperbolic functions.
Texas Instruments has included several applications on the TI-83-Plus calculator. There is a spreadsheet, organizer for phone numbers, puzzle pack, probability simulation, and CBL/CBR applications
The Texas Instruments TI-83-Plus Silver edition calculator has many improvements over the standard TI-83-Plus calculator. The processor is 2.5 times faster, and it has over 1.5 MB of memory – 10 times that of the TI-83 Plus. The improvement in memory makes the TI-83-Plus Silver Edition faster and allows you to store more equations and programs.
The screen on the TI-83-Plus silver edition is very large LCD display that has 64 x 96 pixel resolution. The display can show eight lines of 16 characters each.
A graph link cable can be used to download applications that are specific to different subjects such as physics and geometry. You can use the graph link cable to download applications, games, data and more.
This calculator is also approved to be used on the SAT and ACT college entrance exams.
Every math student in high school or college needs a graphing calculator. A graphing calculator will allow you to easily solve complex mathematical equations. These equations can then easily be graphed on the calculator. You can zoom in to any point on the graph or be able to trace the graph itself.
The Texas Instruments Ti-nspire graphing calculator is clearly a revolutionary idea in the world of graphing calculators. Kids today are well versed in how to use a computer including how to use drop down menus, click and drag interfaces, mouse, documents and folders. The Texas Instruments Ti-nspire graphing calculator brings all those features to this graphing calculator.
Features
·Easy glide touchpad
·High resolution grayscale display
·20 MB Flash Rom
·USB connectivity
·View multiple representations of a problem on a single screen
·Create, save and review work in electronic documents
·Grab a graphed function and move it to see the effect
·"Link" representations
Most students find it difficult to use a graphing calculator and most of the time you see other students taking time to show them how to use their own calculator. The Texas Instruments Ti-nspire graphing calculator is radically different. If you have ever used a computer you will instinctively know how to use the Ti-inspire graphing calculator. The Ti-nspire graphing calculator is an in-your-hand computer used to learn math.
The Ti-nspire graphing calculator allows the student to see what impact a change in the equation will have on the results. You can use the touchpad to move a graph and easily see how this will change the answer.
There is a "linking" feature built into the Ti-nspire graphing calculator. You can make a change in one item in either the equation or the graph and instantly see how it would impact the other. This allows the students to easily understand the relationship.
The Ti-inspire can be used to easily "crack" the SAT or ACT exams. This calculator has been approved to be used on both exams and you will see many students carrying the Ti-nspire graphing calculator with them to take these exams.
Texas Instruments includes a 1 year warranty.
You can easily add a free TI-84 keyboard to this calculator. The only problem is that you must either mail in a card to Texas Instruments or request it online and then they will send it to you. It would be better if this was actually included with the product.
A graphing calculator is a learning tool designed to help students visualize and better understand concepts in mathematics and science. It allows students to make real-world connections in a variety of subjects. As they gain a deeper understanding of the material, students acquire the critical thinking and problem-solving skills they need to attain greater academic success.
Without argument, the graphing calculator reduces the time and effort required to perform cumbersome mathematical tasks. The Texas Instruments TI-89 Titanium Graphing Calculator is capable of providing multiple representations of mathematical concepts.
Features
·Motorola 68000 32-bit microprocessor running
·Algebraic factoring of expressions
·Algebraic simplification (CAS)
·Evaluation of trigonometric expressions to exact values
·Equation solving for a certain variable (CAS)
·Finding limits of functions
·Symbolic differentiation and integration
·Directly programmable in TI-BASIC
·Can run third-party applications
·Large 100 x 160 pixel display for split-screen views
·188 KB RAM and 2.7 MB flash memory for speed
Flash Technology
Most calculators are static. If new technology becomes available in a future product, you had to buy the new product to get it. The TI-89 titanium graphing calculator has 2.7 MB of flash memory. This allows you to upgrade to the latest software version without having to invest in buying a new calculator.
The TI-89 has 188K of RAM. This is more than enough RAM to be able to store functions, programs and data created by the user. The combination of the 188K of RAM and 2.7MB of flash memory allow the TI-89 to be very responsive in completing calculations.
Program Editor
Texas Instruments has included a program editor on the TI-89 titanium graphing calculator that lets you write custom applications. By building tables, tracing along curves, and zooming in on critical points, students may be able to process information in a more varied and meaningful way.
Computer Algebra System
The Texas Instruments TI-89 titanium graphing calculator contains a computer algebra system or CAS. The CAS feature on the TI-89 allows you to solve equations both for a numerical solution like x=4, but also for an answer where you want x in terms of y (like y=2x+3).
Large Display
The TI-89 graphing calculator has a very large 100 x 160 pixel display. You can actually display data in a split screen mode if you want. You can zoom in on graphs to look at an individual point. The display on the TI-89 is LCD. The display can be adjusted so that it can be easily viewable in any lighting condition.
Hard Shell Case
The TI-89 graphing calculator comes with a hard shell case. The cover can be flipped off and attached to the bottom when in use. When not in use, the cover easily protects the TI-89 from abuse in a backpack or if it is accidently dropped.
USB Connection
The TI-89 includes an USB connection to allow you to connect to your PC. You can also use the input/output port to sync with other TI-89 calculators.
Many high school students will have a graphing calculator that they use in Algebra and possibly pre-calculus when they are in high school. When they go on to college they soon discover that their graphing calculator cannot handle the requirements for college mathematics, science or engineering classes.
The Texas Instruments TI-86 graphing calculator includes a function evaluation table, deep entry recall, seven different graph styles with multiple line and shading options, and slope and direction fields for differential equations. The TI-86 can show numeric output for function, polar, parametric, and differential equation modes.
Features
·Graphing functions
·128k RAM with 96K user available
·Function evaluation table
·Input / Output port
·Zilog Z80 microprocessor
The TI-86 is a step ahead of its predecessor the TI-85. The TI-86 graphing calculator includes the same matrix features found in its predecessor but includes a new matrix editor. The new editor on the TI-86 allows students to view and edit matrices in two dimensions.
The TI-86 includes a new function evaluation table. The TI-86 graphing calculator can calculate minimums, maximums, integration, derivatives, arcs and roots. If you need to know what points are in a function, enter the function into the graph, and select table. The TI-86 will present you with an x,y table in a nice graphical format. All the points in the equation can be displayed in the table.
The screen on the TI -86 64 x 128 pixels which can display eight lines of 21 characters. There are two levels of display menus. It is high contrast so it is very readable in different lighting conditions.
The TI-86 graphing calculator is programmable. It was introduced in 1997. The fact that this calculator is still very popular 15 years after its introduction is a testimony to its power, features and ease of use. Over 80% of the product reviews for this product rate it as either 4 or 5 stars.
The TI-86 calculator runs on 4 alkaline batteries. You will find that you will easily drain the life out of the batteries. I would recommend getting four 750mAH NiMH batteries because they will last longer than the alkaline batteries.
The only negative is that the statistics package that is found on the TI-83 is not automatically included with the TI-86. You can easily download it though from and install it using the link cable.
Any student in an advanced math class in high school or calculus class in College will need a graphing calculator. A graphing calculator will allow you to solve an equation and then graph the results. A person is better able to learn the material when they are able to visually see the answer than by looking at an equation.
Texas Instruments has been making precision equipment for decades. The TI-84 Plus graphing calculator has become a common sight around both high school and college campuses due to the combination of its feature sets and its price.
Features
·24Kb of RAM
·480KB of Flash ROM
·13 pre loaded applications included
·8 lines by 16 character display
·Trace graphs
·Kickstand slide case
·USB connectivity
The predecessor to the TI-84 Plus was the TI-83 Plus which might be the best selling calculator ever. The TI-84 Plus graphing calculator builds on the success of its predecessor by offering more features. You can work mathematical calculations, graph them and easily understand the answer with the TI-84 Plus graphing calculator.
The TI-84 Plus graphing calculator runs off of 4 AAA alkaline batteries and one silver oxide battery. The silver oxide battery is used for backup. The 4 AAA alkaline batteries are not included so I recommending you purchase AAA batteries with the TI-84 Plus graphing calculator so you can use it as soon as you receive it. The TI-84 has an automatic shutoff to conserve battery life.
The TI-84 Plus graphing calculator has a very unique kickstand flip case. Most calculators have a flip case. The TI-84 Plus goes a step further to include a kickstand that holds the calculator upright so that you can easily read it while in use. The TI-84 Plus case is very durable which is very important because school age kids will abuse the calculator and the case will stand up to this abuse. It can easily be thrown into a backpack and dropped on the ground and not be damaged.
The included USB cable with the TI-84 Plus makes it very easy to transfer data. The calculator already comes with 11 included applications. You can install new applications on your TI-84 Plus by downloading them from your computer. Some of the preloaded applications include Transformation graphing, Conic Graphing, Inequality graphing, Topics in Algebra chapter 1, chapter 2, chapter 3, chapter 4, and chapter 5. Cabri, CBL, StudyCards, Science Tools, simultaneous probability, and TimeSpan are the other applications that you will notice in your new TI 84 Plus graphing calculator.
The biggest complaint against this calculator is that is very large when compared to its competitors. You could get a smaller calculator but you would get fewer features and pay more for it.
generally skyrocket in the last few years with more challenging textbooks to purchase and additional after school activities. <a href='In this way, both students will benefit from additional stimulation and extra socialization.
Are you searching for the right graphing calculator? If so the Texas Instruments TI-84 Plus Graphing Calculator has everything that you're looking for. Why purchase the TI-84 Plus? It's an awesome unit and handles everything from calculus, engineering, trigonometric, and also financial functions plus unlike other graphing calculators the Texas Instruments TI-84 comes with USB on-the-go technology for file sharing with other calculators. You can connect this calculator to your PC as well. The TI-84 plus is great for complex math and statistics, because it can display answers in the form of graphs. If you're a parent you should consider getting this calculator for your student because it will help them to successfully solve their mathematics and science material. Students can easily share their work on the TI-84 Plus because the built in USB port makes data transfer to computers and between hand held's very easy.They are also able to perform all the complex algebraic or geometric calculations of the most expensive and high-tech calculators on the market, and they can do it all for free. Free online calculators can also be found very easily. In addition to a simple Google search, any number of math blogs on the Web contain link to some great online calculators that are sure to serve whatever purpose you may have for them. While it is true that the best option for someone who has the money and scientific or mathematical knowledge would be to purchase an expensive and sophisticated calculator for the store, those of us who just need to make a quick calculation or conversion can do well with one of the many free calculators that can be found online.
Applying for a mortgage loan is a huge financial and emotional decision that needs to be taken with utmost concern and understanding and the monthly repayment is again the biggest outlay of every month especially when you will see that you are biting off more than you can chew. If the venture is not affordable then the payment of each month to repay the loan becomes a huge responsibility.
Thus to get complete relaxation it is important to seek advises from an expert so that they plan out the best type of loan for you with the minimum interest rates. These mortgage brokers have various options to make life simple and easy and one of the options which are readily available is the online mortgage calculator. Before the advent of the Internet the calculation related to loan were done by loan specialist or accountants and borrowers often had confusion in understanding the concept and the calculations involved in it, but as the online system is considered to be a boon these days so definitely the online mortgage calculator is also a big relief to the borrowers.Different annuity calculator takes different rates of inflation to work out the actual buying power of your estimated pension income at the time of your retirement. In most of the annuity calculator four variables are used and values of the three variables are to be filled up by you and the calculator works out the value of the unknown variable. As a whole, annuity calculator is an extremely useful tool to let you know how accurately your present investments can be fitted into tomorrow's world.
The internet is a booming marketplace. Online automotive lending is an industry that has begun to boom. There are several benefits of getting an automobile loan online, but there are some tips you should follow to fully utilize those benefits.
Online Credit Score The internet is a quick and hassle free place where you can purchase goods/services and acquire useful information.This calculator can prevent you from falling victim to this type of scam.
Compare quotes The internet provides a perfect venue for you to quickly and efficiently compare auto lender quotes. A useful tip for comparing is to use online sites that encourage lenders to compete for your business. This competition leads to lower interest rates and possibly shorter auto loan terms.
The internet is a great resource for individuals looking for an auto loan. If online features, such as credit scores, payment calculators, and competition sites, are used to their fullest, the borrower will always win.
Multi-Level Marketing (MLM), also known as Network MarketingThere are two types:Subscription or Money Plans and Product Based MLM. Subscription or Money PlansSubscription or Money plans are generally illegal and are not really Network Marketing or MLM. False credibilityThe promoters often call them that to give themselves some credibility They generally work by getting you to pay a subscription. Most of which the person or company running the plan gets and pay a percentage out to the person who introduced you to the plan.
You then go out and recruit as many people as you can so that you will get as much commission as possible. There is generally no product or service offered, or if there is it is generally worthless and just incidental. These plans are illegal, although are still often appearing on the doormat, coming in from countries out with the UK.RegistrationTo become part of it's sales force you have to become what they usually refer to as an agent or distributor. For this privilege it will normally cost anything between 45 and 200. 200 is currently the maximum set down by the DSA. This will normally provide you with a company training manual and operating procedures and a few samples of sales brochures, DVD's and stationery.
Now the general idea of network marketing is to get a lot of people retailing a little or in the case of one of the most well known organisations, getting a lot of people to retail a little but also to use all the consumables themselves. It is also sometimes possible to treat the business purely as a retailing opportunity. In fact for over two years I retailed just one of the product range from one MLM company as my sole source of income.Cash flowOften these companies will also give you credit so that you can order and then pay when the invoice arrives which is hopefully when you have collected your customers payments. If you do not have this facility then get a credit card and you will never have cash flow problems. In fact, I run two credit cards, one solely for business use and one for my own personal use.
Provided that you use a credit card wisely, you actually have extra rights under the 1974 Consumer Credit Act. Provided that you carefully checked the terms when opening your credit card account, then you will never incur expenses if you pay off the full amount before the final 'pay by' date. You should be able to achieve this if you are running your business correctly.Payments on interest-only mortgages, of course, are a lot easier to calculate – involving the multiplication of the amount borrowed, by the number of years, by the interest paid. The mortgage repayment calculator really comes into its own, of course, when you have some serious decisions to make about your mortgage. If it is your first, then you will want to know down to the last penny just how much the monthly repayments will be for the interest rate you are quoted. You may also probably want to compare the shorter- and longer-term costs of a repayment mortgage against an interest only mortgage. The calculator will help you compare the offers available from competing mortgage lenders. If you already have a mortgage, you might be interested in the effects of any rise or reduction in interest rate.
Burlington
BurlingtonEven KT our highest Sales $ Sales Rep has half of the customers he calls on earning less than negative $30! DH, while lower in Sales $, has more customers in positive profit territory. Of course some Sales Reps are much worse. Half of DB's customers lose $469 each year. From this data table and graph we can see that high and low performing Sales Reps have many unprofitable customers.
In fact, the best Sales Reps "hide" their bad customers because they have some very good customers. Improvement Action ItemsFrom this data we learned that all Sales Reps have work to do. Everyone has unprofitable customers. If you go back to the formula that we used to calculate profitability, you can understand what the Sales Reps had to do. The driver for these problems is the number of orders.This company's customers place orders too frequently. Instead of ordering a bulk delivery once a month, they have customers ordering barrels two-times a week. The Sales Reps had to go back to their customers and work together to understand the benefit to both companies of reducing order frequency. (If it costs us $105 to process an order, it also costs the customers to process purchase orders and invoices.)This change has allowed the business to grow, adding customers and volume, while reducing the number of people working in order processing.(They are also working on reducing order-processing labor costs through Lean and information technology.
Excel is perhaps the most important computer software program used in the workplace today. That's why so many workers and prospective employees are required to learn Excel in order to enter or remain in the workplace. From the viewpoint of the employer, particularly those in the field of information systems, the use of Excel as an end-user computing tool is essential.
Not only are many business professionals using Excel to perform everyday functional tasks in the workplace, an increasing number of employers rely on Excel for decision support. In general, Excel dominates the spreadsheet product industry with a market share estimated at 90 percent. Excel 2007 has the capacity for spreadsheets of up to a million rows by 16,000 columns, enabling the user to import and work with massive amounts of data and achieve faster calculation performance than ever before.Standard model cosmologists now play that expanding Universe 'film' in reverse. Travel back in time and the Universe is contracting, ever contacting. Alas, where do you stop that contraction? Well the standard model says when the Universe achieves a volume tinier than the tiniest subatomic particle! When (according to some texts) the Universe has achieved infinite density in zero volume – okay, maybe as close to infinite density and as close to zero volume as makes no odds. Translated, in the beginning the Universe was something within the realm of quantum physics! Now just because you can run the clock backwards to such extremes, doesn't mean that that reflects reality. How any scientist can say with a straight face that you can cram the entirety of not only the observable Universe, but the entire Universe (which is quite a bit larger yet again) into the volume smaller than the most fundamental of elementary particles is beyond me.What kind of physics is that? Curiouser and curiouser. Any and all miracles, Biblical or otherwise, are explainable as easily as saying "run program". More down to earth, you have multi-observations of things like the Loch Ness Monster, those highly geometrically complex crop circles, and ghosts, yet there's no real adequate theory, pro or con, that can account for their observed existence or creation. All up, perhaps some cosmic computer programmer/software writer whiz with a wicked sense of humour (a trickster 'god'?) is laughing its tentacles off since we haven't been able to figure it (our virtual reality) out. Of course maybe the minute we do, the fun's over and 'Dr. It' hits the delete key and that's the way the Universe ends – not with a Big Crunch, nor with a Heat Death, but with a "are you sure you want to delete this?
If you're a scientist, you probably don't think you need an iPhone app to help you with your endeavors, but many may actually help. And, if you are an aspiring scientist or man of science, or if science simply titillates and excites you, a number of iPhone applications were made for you! There are a lot of apps related to science for the iPhone, but a few stand above the rest. Why waste your time (and money) on somewhat lacking iPhone applications when you can get the best? Here are the top iPhone applications for scientists and those who want to unleash the scientist in them:JottNo scientist can conduct an experiment or study without a recording tool. This app turns your iPhone into a recorder. It actually records your voice and turns it into text. Talk about a hands-free way to create a pile of notes! Grafly and SolutionsThese two iPhone applications are so useful that many scientists use them and have raved about them.Since there is no multiple answers the children cannot guess. Kid Calc Elementary Math Help with Flash CardsThis math app is actually four apps all put together. It focuses on preschool and elementary students. It uses cool graphics along with fun games to keep young children engaged. There are animated flash cards, counting games, addition and subtraction math drills, and an animated calculator.
Ben Spratling Math TouchIf you are in a science class or an engineering student, this is considered the best math app. It automatically converts units and vector coordinate systems. It has an astronomical observation database and an equation database. Math Ref FreeSome math formulas are hard to remember but Math Ref Free takes the guessing out of it, helping you to understand about a particular formula and to find formulas.
There are also all kinds of tips to make understanding the formulas easier. Free Graphing CalculatorThe iPhone comes with different calculators to help you with your calculations. The free graphing calculator is a very powerful math app and it's also very flexible.Memory is usually allowed on graphing calculators by the organizations administering standardized tests. However, students with memory-enabled calculators are not allowed to bring stored examples into the exam, or take out the exam questions afterwards. This means that the memory must be cleared both before and after the exam. Calculator Tips for Exams:* Bring your calculator, even if you may not need it, they are not usually available at test centres.* Practice using your calculator on sample SAT mathematics questions before the test.* Don't buy an expensive, sophisticated calculator just to take the test – the problems simply do not require it.* Don't try to use a calculator on every question. First, decide how to solve the problem, and then decide whether to use the calculator. The calculator is meant to help, not get in the way.For those looking for a more standard scientific calculator, Powerone LE fits the bill. It's got a very intuitive interface that you'll pick up right away. Powereone also features unit conversion and a currency converter that stays current to international currency exchange rates. Additionally, its got a simple statistics calculator that finds the mean of a series of numbers. It could be better if the scientific features used a two-line interface.
GRAB A BUSTY BEER WITH FIND ME A GIFT'S SEXY BEER BOTTLE HOLDER! Find Me a Gift lets you slip your hands around a womanly waist without the fear of a slap! Findmeagift. com is an online gift company based in the Midlands. They stock a bevy of gifts, gadgets, gizmos and gift experience packages. You can be sure to get your hands around something unusual! The Sexy Beer Bottle Holder is a bottle holder in the shape of a sexy, bikini-clad body! The colours do vary, and the titillating two-piece fashioned holder is available in Red, Pink, Black and White. The Sexy Beer Bottle Holder measures approximately 14 cm x 10.5 cm x 9 cm so will snuggle nicely around any standard-sized beer bottle! After a hard day at work or relaxing on a summer's evening, you can't beat an ice-cold beer!When others raise a glass, you can lift the Sexy Beer Bottle Holder up high too, in true Richard Gere style! So hold on tight, don't let this beauty slip through your fingers!
So whether you are looking for a naughty birthday gift, a fun stag night accessory or simply for a novel sleeve to slip your beer into, the Sexy Beer Bottle Holders are just what the barman ordered. Forget women shaped like 'hour glasses'! The Sexy Beer Bottle Holders are making 'beer bottled' shaped women hotter. For a super sexy and ice cool way to sup your favourite bottled beverage, get your hands around one now! Find Me a Gift offers everything online without the need for people to spend money on petrol, parking and inflated prices.They have screens that can be navigated by touchpads, the ability to save documents, and have such a high resolution that users can upload pictures. My graphing calculator review is that many schools think that these calculators are a little too advanced. Some of these calculators can even connect to the Internet. This makes it much easier for students to cheat on tests.
Hence, schools continue to make students to purchase the old TI-83 and TI-84′s – calculators that are more expensive, and thirteen years older than some of the more advanced, contemporary calculators. While writing this graphing calculator review, it occurred to me that all of these models (including the TI-83 and 84) are great learning tools that can make math easier and fun.As well, tips for college visits before, during, and after the application process will be provided.
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This course develops students' ability to recognize, represent, and solve problems involving relations among quantitative variables. Key functions studied are linear, exponential, power and periodic functions using graphic, numeric and symbolic representations. Students will also develop the ability to analyze data, to recognize and measure variation and to understand the patterns that underlie probabilistic situations
This course prepares students to take the Advanced Placement Calculus Exam. It covers limits, differential calculus, and integral calculus. Terminology, theory, notation, and in-depth problems guide students through a rigorous study of calculus. Extensive use of the graphing calculator will be involved throughout the year.
Students will review and extend their knowledge of linear, quadratic, exponenetial, logarithmic, polynomial, trigonometric, step, and absolute value functions. Students will apply elements of probability, theory, and concepts of statistical design to interpret statistical findings. Other skills emphasized include imporving arithmetic skills, algebraic maniputlation, solving equations without calculators, and solving systems of equations.
This course is designed to prepare students with tools necessary to be successful when taking the MCA II. Students will learn test-taking skills and be exposed to the MCA II testing format in addition to learning how to use the TI-83 graphing calculator. Students will work on practice tests and study key components of the MCA II exam.
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Theory and Problems of Elementary Algebra
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"This third edition of the perennial bestseller defines the recent changes in how the discipline is taught and introduces a new perspective on the discipline. New material in this third edition includes: A modernized section on trigonometry An introduction to mathematical modeling Instruction in use of the graphing calculator 2,000 solved problems 3,000 supplementary practice problems and more "
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Course Description: Designed
for those who want to improve their attitude toward mathematics.
Explores feelings & develops strategies to overcome math phobia.
Emphasis will be placed on problem-solving approaches to diagrammed,
descriptive, & symbolic number problems. This course is open to
students at all levels of mathematical skills, whether preparing for a
job, college courses, a test, or living in a world where numbers matter.
One hour lecture/discussion each week.
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Math
STEM Mathematics is a study of mathematical methods that are typically used in science, engineering, business, and industry with a review of essential concepts from Algebra 2 and Pre-Calculus. Students will work on application problems which include trigonometry, systems of linear equations, quadratic, exponential, and logarithmic function basics to mathematically model problems and derive solutions with emphasis on the use of technology and other tools.
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Ankit1010 wrote:All right, Real Analysis is a proof-based course. If you have not had one of those before, it will be quite different from your other math classes. No longer will the professor tell you how to solve problems and then ask you to solve them. Now you will have to prove that the techniques to solve those problems are actually valid. You should be familiar with proof by induction, proof by contradiction, and proof by contraposition.
The course will start out covering thing that you've known since elementary school, but have probably never studied rigorously before. For example, in the section about field axioms, you might be asked to formally prove that 0·1=0. You might have to rigorously define what it means for a real number to be "positive", and prove that the positive numbers are closed under addition, multiplication, and division, but not subtraction.
From the list of topics you provide, it seems likely your course will not make you prove that such a thing as the real numbers actually exists. Instead it will implicitly take the stance, "If a complete ordered Archimedean field exists, these are the properties it must have."
Most of the course will stem from a few major ideas. One of these is the least upper bound property, meaning every bounded set of real numbers has a least upper bound in the reals. Another is trichotomy, meaning for any two real numbers x and y, exactly one of "x=y", "x<y", and "x>y" is true.
There will be a lot of topics dealing with limits from the ε-δ definition. This will probably be couched in the language of neighborhoods and balls, and likely will constitute your first introduction to the study of metric spaces. The course will start slowly, then progress quickly, until at the end you will be rigorously proving calculus theorems that you might not have seen before.
Qaanol wrote:And it really is a ton of fun. If you like that sort of thing, anyways.
Ben-oni wrote:Practice proofs. Look up any of the terms listed that your not familiar with. That should do you good for now.
Qaanol wrote:... AsI've actually done several proof-based courses before and am very comfortable with writing and understanding formal proofs. Let me clarify what I meant in the question - I want to know what textbooks/problem sets/video lectures/resources I can use to start learning the material that will be taught DURING the class, not the general areas I should cover for background knowledge. The fact is that my GPA that needs a lot of work, and next semester promises to be tough so I'm trying to effectively teach myself everything we do in class over the summer to get a head-start.
Understanding Analysis by Stephen Abbot is a very good book for self studies and the toc is suspiciously close to your course description. A good textbook in combination with wikipedia and will probably do the trick.
Alright, thanks for the advice! Understanding Analysis looks like a great book, I'll definitely pick up a copy, and I'll find out which text we'll be using for the class soon too. It will likely be either Principles of Mathematical Analysis by Rudin or Understanding Analysis. I love stack exchange, and that coupled with this forum for more serious difficulties should be enough to resolve any problems.
Ankit1010 wrote:I'm taking undergrad Real Analysis I at college next semester, and I want some advice on how I can start preparing for the course over summer since I need to do well on it.
I had a wonderful academic experience in that course. There were two factors I've identified.
1) I had a fabulous teacher. You have some control over that if there's more than one section and you can find reviews. This is difficult material ... and for someone who wants to do advanced math or physics, it's absolutely essential to nail this course. So get the best teacher you can.
And along these lines ... buddy up to the TA's. The TA's are grad students who still remember what it's like to not understand this material ... so they can be incredibly helpful. Join a study group. Go to TA office hours. Be friendly. Hang around with the other students. It really helps to grapple with this material with other people.
2) I took it during summer school and took nothing else. In fact I was taking an upper division computer science class and just dropped it. I did nothing but real analysis. And it really made a difference. This is a very labor-intensive course. You just have to do epsilon proofs till they come out your ears. Because the course involves concepts that are deep; and techniques that are precise. You really have to put some time into this class.
That would be my advice. Sign up with a good prof; hang out with the TA'S and other students; and clear the decks in the rest of your life so that you can spend all your time thinking about real analysis.
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About Business Math: This class is designed for Seniors. This is a one year course. Business math can count toward MATH or ELECTIVE Credits (5 per semester). Students will cover the essentials as well as the basics in math. Students learn about real-life scenarios such as Managing Money, Loans, Payments, Interest Rates, Taxes, Insurance, Purchasing an Auto/Home, etc. There is an extensive amount of basic skills review (fractions, decimals, percents, order of operations, etc). A basic four-function calculator is required. Students MAY NOT use cell phones in class and may not enter classroom without valid SFHS ID.
LATE OR INCOMPLETE ASSIGNMENTS AND TESTS MUST BE MADE UP WITHIN 2 DAYS OF DUE DATE (WITH PARENT PHONE CALL); WARM UPS ARE GIVEN DURING THE FIRST 10 MINUTES OF CLASS AND MAY NOT BE MADE UP.
About Guided Study: This class/program is designed for freshman and/or sophomores (selected by SFHS). Students receive a Classroom, a Teacher, Mentors, Resources, and CREDITS to do assignments. Behavior, attendance, and grades are monitored very carefully by the Teacher and the Guided Study Program. EXTREMELY IMPORTANT: ALL STUDENTS MUST BRING ALL MATERIALS FOR ALL CLASSES EVERY DAY (EVEN ON BLOCK DAYS). Required materials: 2+inch 3-ring binder, dividers, lined paper, pencils, pens, School ID.
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For Students
'Learning math ultimately comes down to one thing: the ability, and the choice to put one's brain around a problem, stare past the confusion and struggle rather than flee.'
(Sept. 2002 issue of Mathematics in the Middle School)
Annual Integration Bee
The 2013 Integration Bee is planned for Friday, May 10, in Room 2625, starting at 5 p.m. View the 2013 official rules.
This year's prizes are good ones! First prize is $100 cash. Second prize is $75 cash. Third prize is a $50 cash. Fourth prize is $25 cash.
Past Winners
2010
Miguel Corona - TI-89 Graphing Calculator
2006
Hari Tyagi - $150 Cash
2005
Nicole Petreust - IPOD shuffle
2004
Jeff Rade - Palm Zire 71 PDA
2003
Brian Snell - I-89 Graphing Calculator
Sponsors
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2004
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2003
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Addison Wesley
Many schools have integration bees. Although we have not published all of the integrals that we have used in previous years, some schools have. For practice integrals, search the web for "integration bee" or consult any calculus textbook. Here is one list of practice problems that may be useful: University of North Texas Integration Bee.
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I am only a first year graduate student, but I am very interested in mathematics education. My own approach to teaching is very much problem based: give students interesting problems which lead to the development of the concepts you want them to have. Even if they can't come up with all of the needed concepts on their own, if you give it to them after they have wrestled with a problem they will be much more likely to be able to apply the concept in novel situations in the future. Why couldn't this approach be carried through in a math grad situation? Design a sequence of problems, varying in difficulty, which in total cover need material from most of the "first year curriculum".
Before writing this off as a crazy idea, I would like to point out Cornell's vet school. They use exactly the model given above: Every week or two there is a new case. In each of your classes (anatomy, pharmacology, radiology, ...etc) you cover general information which is pertinent to the case of the week, but it is up to you and your team to do research, come up with a diagnosis and a method of treatment. So all of the classes you take are integrated together in the context of solving some real problems. Cornell is turning out some amazing vets. Why couldn't the same model work for mathematics?
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This book contains a large amount of information not found in standard textbooks. Written for the advanced undergraduate/beginning graduate student, it combines the modern mathematical standards of numerical analysis with an understanding of the needs of the computer scientist working on practical applications. Among its many particular features are: (bullet) fully worked-out examples (bullet) many carefully selected and formulated problems (bullet) fast Fourier transform methods (bullet) a thorough discussion of some important minimization methods (bullet) solution of stiff or implicit ordinary differential equations and of differential algebraic systems (bullet) modern shooting techniques for solving two-point boundary value problems (bullet) basics of multigrid
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General: We will follow the course outline of Prof. Richard Froese's class in previous term, which you can download from here: lecturenotes. However, the detailed topics may differ as will the homework and grading (we have no text book). Related information for section 201 of this course (by Prof. Joel Friedman) can be found here. Some simple programming is needed for this couse in Matlab or Octave. A page on Matlab/Octave in the math department is available here. For documentation you might try: MATLAB documentation page and GNU Octave page. GNU Octave is free, and is (usually) not too difficult to install on any Linux, MacOS, or Windows system (Linux systems usually come with Octave installed). Richard Froese has created a UBC Wiki page to help you install Octave if you'd like to give this a try; see also the GNU Octave download instructions.
Objective: We learn basic knowledge in linear algebra first (theoretically), together with programming using Matlab/Octave (which is suitable for matrix-matrix computation); during learning, related problem about practical application will be inserted, such as interpolation, finite difference approximation, power method, network connectivity, recursion relations, Fourier transform, etc.
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Hello there, I'm a student in high school and I'm tormented by my assignments. One of my issues is addressing online textbook mcdougal littell; Will someone on the web help me in understanding what it's all about? I need to complete this immediately! Thanks to allfor assisting.
I think I understand what you are searching) for. Try out Algebra Buster. This is a wonderful tool that helps you get your assignments completed quicker as well as correct. It can help out with courses in online textbook mcdougal littell, inequalities plus more.
Registered: 24.10.2003
From: Where the trout streams flow and the air is nice
Posted: Sunday 21st of Nov 18:29
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Information
Contact Information
Tutorials
JoAnne Lee
Welcome to 8th grade Algebra I Pre-AP!
The overarching goal of Holt Algebra I is to help students develop mathematical knowledge, understanding, and skills, as well as awareness and appreciation of the rich connections among mathematical strands and between mathematics and other disciplines.
*There has been a change to the 8th grade Homework Hall policy. If a student fails one or more of his/her core classes, he/she will lose the option of going to Homework Hall and be required to attend a mandatory Lunch Homework Hall.
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The District 59 junior high mathematics program was chosen to meet the needs of students of the new millennium. The goals of the program are to:
Provide teachers and students with materials that implement the National Council of Teachers of Mathematics (NCTM) curriculum and evaluation standards and the Illinois Learning Standards for Mathematics (ILSM)
Provide students with access to high school mathematics success
Implementing the NCTM and ILSM Standards Our junior high mathematics program incorporates the NCTM and ILSM standards. It is a program with rigorous content that builds on prior knowledge and provides carefully sequenced instruction in the building blocks of algebra: numerical, proportional, and algebraic reasoning.
Course 1, which is covered at the sixth grade level focuses on numerical reasoning and lays a solid foundation for algebra readiness.
Numerical reasoning includes:
Number sense
understanding numbers and their relative magnitudes
rounding numbers
finding patterns
using referents for estimating measures of common objects
recognizing real-life uses of numbers
Operation sense
estimating results of computations using strategies
doing mental computations using strategies
understanding the effects of operations
choosing an appropriate operations/computation method
interpreting the answer
Course 2, which is covered at the seventh grade level focuses on proportional reasoning providing all the readiness skills necessary to prepare students for high school algebra.
Proportional reasoning includes:
recognizing, using, and expressing proportional relationships
finding equivalencies between ratios and percents
using tables and graphs to compare proportional and non-proportional relationships
using proportional reasoning in common applications
Course 3, which is covered at the eighth grade level focuses on algebraic reasoning and provides an extension to Course 1 and 2 for students.
Algebraic reasoning includes:
translating real-world and mathematical relationships into expressions and equations
representing algebraic relationships in tables and graphs
understanding variables and constants
applying formulas
finding patterns in sets of equations
understanding functions and relationships
using algebra to solve problems
The junior high program provides daily real-world problem solving experiences, stresses oral and written mathematical and cross subject communication skills. The program also makes connections between mathematics and other academic areas, provides varied technology (computer and other) connections and applications for developing understanding and presenting mathematical concepts and problems, opportunities for independent and cooperative learning, and provisions for a variety of student needs and learning styles.
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An intensive refresher course in basic mathematics with introductory algebra topics. This course prepares students for MAT 136, Mathematics for the Health Sciences. Topics include fractions, decimals, ratio and proportion, percents, solving linear equations and inequalities, graphing linear equations, and operations and polynomials. College credit will be awarded, but this credit will not count toward a degree. This course is designed to meet the mathematics prerequisite for MAT 136 ONLY and course enrollment is restricted to nursing and health students whose program or program prerequisites require(s) MAT 136. Skills prerequisites: MAT 011, ENG 020 and ENG 060.
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The Advanced Algebra Tutor: Learning By Example DVD Series teaches students through step-by-step example problems that progressively become more difficult. This DVD covers the transformations of functions in Algebra, including what a transformation of a function is and why it is important. Grades 9-College. 35 minutes on DVD.
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In it's purest form it can provide endless riddles and puzzles to solve and provide solutions and answers to some of life's biggest questions. And in practical ways it can help us make the best possib...
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Most of us have experienced the amusement (and possible embarrassment) that goes with standing in front of a distorted funhouse mirror. What many people don't realize is that convex and concave mirror...
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Using exciting live-action demonstrations and easy-to-understand animation, this video delves into the fundamental concepts of reflection and its relationship to light, vision, and the physical world....
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Anyone standing in front of a mirror will instantly recognize the concept of reflection at work, but to observe the process of refraction and to develop an in-depth knowledge of it is quite a differen...
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The mathematical skills children bring with them to elementary school predict both their mathematical and literacy achievement for years to come. In this video, experts from Erikson Institute's Early...
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Instructors who are looking for a way to integrate handheld technology and visual media into their algebra classes will benefit greatly from this ten-part series. In each program, internationally accl...
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What do mathematicians mean when they say that an event has a 50 percent probability of occurring? How does the study of statistics apply algebraic principles to real-world events and conditions in a...
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Most people have a basic understanding that exponential growth means rapid growth. But framing this concept in algebraic terms and applying it to concrete problems in real-world situations is a differ...
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Environmentalists, meteorologists, economists, and people in many other disciplines have always been interested in the dynamics of variables, or quantities that change-for example, the number of polar...
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Minor steps like reversing the "less than" or "greater than" sign might look simple, but when students first try to grasp the strange world of inequalities, they often feel overwhelmed. That's a probl...
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105. Preparation for College Mathematics. Every Fall
Introductory logic and algebra, elementary functions: polynomial, rational, trigonometric, exponential, logarithmic. Prerequisite: Permission of the department. Not for credit toward the mathematics major or minor. Buehrle
109. Calculus I. Every Semester
Introduction to the basic concepts of calculus and their applications. Functions, derivatives and limits; exponential, logarithmic and trigonometric functions; the definite integral and the Fundamental Theorem of Calculus. Prerequisite: Twelfth-grade mathematics or MAT 105. Staff
130. Mathematics of Art. 2013 – 2015
This course uses mathematical perspective to analyze works of visual art (perspective drawing and perspective geometry). The first topics of this course will use one-, two- and three-point perspective 130 both to create realistic pictures and to study optical illusions. From there we will explore other dimensions: what does a four-dimensional cube look like? What does a 1.638-dimensional object look like? The course will finish with fractals and chaos theory, which we will use to draw textures of natural objects such as ferns and clouds. Crannell
211. Introduction to Higher Mathematics. Every Semester
A course designed as a transition from calculus to advanced mathematics courses. Emphasis on developing conjectures, experimentation, writing proofs and generalization. Topics will be chosen from number theory, combinatorics and graph theory, polynomials, sequences and series and dynamical systems, among others. Prerequisite: MAT 111. Staff
370 – 379. Selected Topics.
375. Topics in Algebra. Spring 2014
Courses of an algebraic nature such as Ring Theory, Advanced Linear Algebra and Algebraic Number Theory, that can be taken in place of, or in addition to, MAT 330 to satisfy the major requirements. May be repeated with permission of department. Prerequisite: MAT 211.
390. Independent Study.
Independent study directed by the Mathematics staff. Permission of chairperson.
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This online course presents a conceptually-based approach to teaching mathematics and uses the straight-line coordinate geometry unit in which to do this. The course utilises GeoGebra, however, other dynamic geometry software, or indeed no dynamic geometry software, may be used.
The success of this engaging, conceptual, inquiry-based approach lies in the fact that students, during the first few lessons, learn to perform the following skills without utilising formulae: read gradients from straight lines, sketch straight lines directly from equations (without using tables of values), 'read' an equation directly from its graph and find the midpoints and distance between pairs of points. Students who typically struggle with coordinate geometry tend to achieve far greater success and confidence with this approach.
Towards the end of the unit traditional formulas are derived and then employed.
Accredited PD Hours = 7
This course was better than I expected.
No problems with any downloads and all materials were professionally presented.
I will go away and practice what I have been shown and I am certain that my students will benefit from my experiences.
Thank You Richard
This was a satisfying course, any new way to teach students which increase understanding, interest and participation is all good.
This was very useful for me at the stage where I am in teaching in a new digital environment.
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Introduction:
Symbolic mathematics is popularly know as "computer algebra".
Computer Algebra
The term "Computer Algebra" is used to refer to the use of computers for performing algebraic or symbolic transformations.
This means the manipulation by computers of expressions or formulae formed using names representing variable or unknown quantities and names or special symbols which denote operations or functions applied to these quantities.
Example
A simple example of the kind of transformation which might be performed symbolically by a computer algebra tool is:
(x2 - 1)(1 + x) => x3 + x2 - x - 1
This transformation is known as "expansion".
Though computer algebra tools are distinguished by their ability to perform a wide range of symbolic transformations they are usually also equally competent in purely numerical computation, and many kinds of task (for example integration) can be performed either numerically or symbolically.
It may also be noted that tools supporting "linear algebra" (such as Matlab) are primarily concerned with numerical computations involving vectors or matrices of numbers, and may have limited symbolic capabilities.
Well known general purpose "computer algebra" tools include Axiom, Reduce, MACSYMA, Maple and Mathematica.
MACSYMA:
MACSYMA was the first general purpose computer algebra program.
Its development began in 1968 at the MIT Artificial Intelligence laboratories, and continues through to the present day, now in the hands of Macsyma Inc.
MACSYMA was designed by Carl Engelman, William Martin and Joel Moses, building on their previous experience with Mathlab 68 and other doctoral research work on particular kinds of symbolic problem solving.
An example of this prior work was the SIN program written by Moses to perform symbolic integration.
MACSYMA provided the first comprehensive interactive computing environment for use by mathematicians, scientists and engineers in solving mathematical problems.
It provided a wide range of algebraic transformation capabilities, backed up by an extensive library of numerical facilities and a plotting package.
On the downside, it was developed using LISP to run on large and expensive timesharing computers and was consequently very resource hungry.
MAPLE:
MAPLE was designed for portability and speed, to make Computer Algebra more widely accessible.
The development of MAPLE was begun in 1980 at the University of Waterloo, Canada.
A key design goal was to broaden access to computer algebra software by reducing the computer resources demanded by the software, and by making the software portable across a wide range of computer models.
For this reason, development of MAPLE began with the B language, progressing to the C language when it became available, and involved careful studies of the efficiency of alternative algorithms.
In 1987 MAPLE was incorporated as Waterloo Maple Software (now Waterloo Maple Inc.), and continues to be developed as commercial software.
Its present capabilities are the result of two decades of development and provide:
Numerical Calculation
Symbolic integration and differentiation
Symbolic equation solving
Trigonometric and Exponential/Logarithm Functions
Linear Algebra
Statistics
Two- and three-dimensional Plotting
Animation
Interfaces with the C and Fortran Programming languages
Mathematica:
Mathematica is today's world leading computer algebra tool.
Stephen Wolfram
Mathematica was developed by Stephen Wolfram, a distinguished polymath educated at Eton, Oxford and Caltech who received his PhD in theoretical physics at the age of 20, and whose work is now cited in over 10,000 technical papers.
Aim
Mathematica aims to provide users with a fully comprehensive integrated environment for all kinds of mathematical application, ranging from elementary computations and transformations to large development projects building mathematical models for use in complex engineering projects.
Approach
Through an interactive programming language optimised for this kind of work, users have access to a comprehensive range of mathematical functions including both the well known standard functions such as sin and cos and Mathematica superfunctions such as Solve, Integrate, and Simplify.
Superfunctions
These superfunctions use the full range of lower level capabilities for certain kinds of problem solving, selecting and applying them as required by the problem in hand.
These superfunctions include "Solve" for exact symbolic and "NSolve" for numeric solution of polynomial equations, "Integrate" for symbolic and "Nintegrate" for numerical integration, "DSolve" and "NDSolve" for simultaneous differential equations, and "Simplify" for simplifying mathematical expressions.
Application Libraries
The power of Mathematica and other computer algebra tools is to a large extent due to the extensive libraries which provide their comprehensive knowledge of mathematics and its applications.
The libraries provided by the supplier are supplemented by compendious online libraries of applications made available by the community of users.
History
Wolfram's interest in computer algebra started with the development of SMP a computer algebra system which he began to develop in 1979 and released commercially in 1981.
After a digression to found the field of complex systems Wolfram returned to computer algebra in 1986 and began the development of Mathematica.
He founded Wolfram Research in 1987 to continue the commercial development of Mathematica.
When first released in 1988, Mathematica seized the lead in this field which it has retained to this day, reaching 100,000 users in 1990 and one million in 1995.
Numerical Computation
Numerical computation in Mathematica is conducted to arbitrary precision controlled by the user.
The user selects the precision he requires in the result and Mathematically automatically selects the precision needed in intermediate results to give that precision in the end result.
Mathematical Libraries
The combination in Mathematica's programming language of those features found in other good high level programming languages together with an exhaustive knowledge of mathematics makes for outstanding ease in implementing mathematical applications.
Alongside these comprehensive mathematical capabilities mathematica provides powerful graphical and multimedia capabilities.
All of this is supported in a document oriented paradigm in which work is prepared and presented in Mathematica "notebooks", suitable either for electronic (fully animated viewing) or hard copy presentation.
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Middle School Math
Through our unique problem solving approach, your child will develop a deeper understanding of pre-algebra concepts so they can better apply what they've learned at school and in the real world. They will develop the flexible, abstract thinking needed to be successful in algebra and higher level mathematics.
Students seeking to improve their problem solving skills and advance to higher math will find our Algebra class both challenging and rewarding. This year-round course covers material in a traditional Algebra I course including graphing and solving equations, factoring polynomials, exponents and logarithms. Our classes will guide students through hands-on and computer based experiences to help them develop a deeper understanding of Algebra.
The creative process involved in developing a formal proof is an important one and this class takes a student well beyond what they would experience in a traditional classroom. The class will strengthen your child's foundation in geometry and develop their logic and reasoning skills as they learn to formulate rigorous mathematical proofs.
The process of formulating proofs allows a student to experience the heart of mathematics and the act of going through the process helps to instill a higher level of learning. From week to week students will be provided with well-known theorems and will be required to complete a proof of the given theorem. The majority of the work is done in-class with the intent to facilitate and supplement the student's learning process. The "out-of-class" assignments consist of geometric constructions and investigation of these constructions. Student may be required to have access to a computer outside of class.
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what do you want to learn?
Algebra Questions
Virtual Nerd teaches you how to answer your Algebra questions.
Get step-by-step instruction for your algebra questions with Virtual Nerd, the source for interactive video tutorials in math and science for students in grades 7-12.
Learn at your own pace.
Use the site whenever you want.
Collaborate with your parents and teachers to track your progress.
And best of all, access hundreds of interactive video lessons to master the fundamentals of key algebra and math concepts.
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